Eternal inflation is an inflationary universe model, which is itself an outgrowth or extension of the Big Bang theory. In theories of eternal inflation, the inflationary phase of the universe's expansion lasts forever in at least some regions of the universe. Because these regions expand exponentially rapidly, most of the volume of the universe at any given time is inflating. All models of eternal inflation produce an infinite multiverse, typically a fractal. The eternal nature of new inflation was discovered independently by Paul Steinhardt and Alexander Vilenkin in 1983.
Eternal inflation is predicted by many different models of cosmic inflation. MIT professor Alan H. Guth proposed an inflation model involving a "false vacuum" phase with positive vacuum energy. Parts of the universe in that phase inflate, and only occasionally decay to lower-energy, non-inflating phases or the ground state. In chaotic inflation, proposed by physicist Andrei Linde, the peaks in the evolution of a scalar field (determining the energy of the vacuum) correspond to regions of rapid inflation which dominate. Chaotic inflation usually eternally inflates, since the expansions of the inflationary peaks exhibit positive feedback and come to dominate the large-scale dynamics of the universe.
Alan Guth's 2007 paper, "Eternal inflation and its implications", details what is now known on the subject, and demonstrates that this particular flavor of inflationary universe theory is relatively current, or is still considered viable, more than 20 years after its inception. 
Inflation and the multiverse
It's hard to build models of inflation that don't lead to a multiverse. It's not impossible, so I think there's still certainly research that needs to be done. But most models of inflation do lead to a multiverse, and evidence for inflation will be pushing us in the direction of taking [the idea of a] multiverse seriously. Alan Guth
It's possible to invent models of inflation that do not allow [a] multiverse, but it's difficult. Every experiment that brings better credence to inflationary theory brings us much closer to hints that the multiverse is real. Andrei Linde 
Polarization in the cosmic microwave background radiation suggests inflationary models for the early universe are more likely but confirmation is needed.
Inflation, or the inflationary universe theory, was developed as a way to overcome the few remaining problems with what was otherwise considered a successful theory of cosmology, the Big Bang model. Although Alexei Starobinsky of the L.D. Landau Institute of Theoretical Physics in Moscow developed the first realistic inflation theory in 1979 he did not articulate its relevance to modern cosmological problems.
In 1979, Alan Guth developed an inflationary model independently, which offered a mechanism for inflation to begin: the decay of a so-called false vacuum into "bubbles" of "true vacuum" that expanded at the speed of light. Guth coined the term "inflation", and he was the first to discuss the theory with other scientists worldwide. But this formulation was problematic, as there was no consistent way to bring an end to the inflationary epoch and end up with the isotropic, homogeneous universe observed today. (See False vacuum: Development of theories). In 1982, this "graceful exit problem" was solved by Andrei Linde in the new inflationary scenario. A few months later, the same result was also obtained by Andreas Albrecht and Paul J. Steinhardt.
In 1986, Linde published an alternative model of inflation that also reproduced the same successes of new inflation entitled "Eternally Existing Self-Reproducing Chaotic Inflationary Universe", which provides a detailed description of what has become known as the Chaotic Inflation theory or eternal inflation. The Chaotic Inflation theory is in some ways similar to Fred Hoyle’s steady state theory, as it employs the concept of a universe that is eternally existing, and thus does not require a unique beginning or an ultimate end of the cosmos.
Quantum fluctuations of the inflation field
Chaotic Inflation theory models quantum fluctuations in the rate of inflation. Those regions with a higher rate of inflation expand faster and dominate the universe, despite the natural tendency of inflation to end in other regions. This allows inflation to continue forever, to produce future-eternal inflation.
Within the framework of established knowledge of physics and cosmology, our universe could be one of many in a super-universe or multiverse. Linde (1990, 1994) has proposed that a background space-time "foam" empty of matter and radiation will experience local quantum fluctuations in curvature, forming many bubbles of false vacuum that individually inflate into mini-universes with random characteristics. Each universe within the multiverse can have a different set of constants and physical laws. Some might have life of a form different from ours; others might have no life at all or something even more complex or so different that we cannot even imagine it. Obviously we are in one of those universes with life.
A 2014 paper by Kohli and Haslam  analyzed Linde's chaotic inflation theory in which the quantum fluctuations are modeled as Gaussian white noise. They showed that in this popular scenario, eternal inflation in fact cannot be eternal, and the random noise leads to spacetime being filled with singularities. This was demonstrated by showing that solutions to the Einstein field equations diverge in a finite time. Their paper therefore concluded that the theory of eternal inflation based on random quantum fluctuations would not be a viable theory, and the resulting existence of a multiverse is "still very much an open question that will require much deeper investigation".
In standard inflation, inflationary expansion occurred while the universe was in a false vacuum state, halting when the universe decayed to a true vacuum state and became a general and inclusive phenomenon with homogeneity throughout, yielding a single expanding universe which is "our general reality" wherein the laws of physics are consistent throughout. In this case, the physical laws "just happen" to be compatible with the evolution of life.
The bubble universe model proposes that different regions of this inflationary universe (termed a multiverse) decayed to a true vacuum state at different times, with decaying regions corresponding to "sub"- universes not in causal contact with each other and existing in discrete regions that are subject to truly random "selection", determining each region's components based upon the persistence of the quantum components within that region. The end result will be a finite number of universes with physical laws consistent within each region of spacetime.
False vacuum and true vacuum
Variants of the bubble universe model postulate multiple false vacuum states, which result in lower-energy false-vacuum "progeny" universes spawned, which in turn produce true vacuum state progeny universes within themselves.
Evidence from the fluctuation level in our universe
New inflation does not produce a perfectly symmetric universe; tiny quantum fluctuations in the inflaton are created. These tiny fluctuations form the primordial seeds for all structure created in the later universe. These fluctuations were first calculated by Viatcheslav Mukhanov and G. V. Chibisov in the Soviet Union in analyzing Starobinsky's similar model. In the context of inflation, they were worked out independently of the work of Mukhanov and Chibisov at the three-week 1982 Nuffield Workshop on the Very Early Universe at Cambridge University. The fluctuations were calculated by four groups working separately over the course of the workshop: Stephen Hawking; Starobinsky; Guth and So-Young Pi; and James M. Bardeen, Paul Steinhardt and Michael Turner.
The fact that these models are consistent with WMAP data adds weight to the idea that the universe could be created in such a way. As a result, many physicists in the field agree it is possible, but needs further support to be accepted.
- Guth, Alan H. (2000). "Inflation and Eternal Inflation". Phys.Rept. 333: 11. arXiv:astro-ph/0002156. Bibcode:2000PhR...333..555G. doi:10.1016/S0370-1573(00)00037-5.
- Guth, Alan; Eternal inflation and its implications arXiv:hep-th/0702178
- Holt, Jim. "The Big Lab Experiment. Was our universe created by design?". Slate.
- Jones, Douglas S. "Many worlds interpretation".
- Guth, Alan. "Eternal inflation: Successes and questions".
- Our Universe May Exist in a Multiverse, Cosmic Inflation Discovery Suggests
- Starobinsky, A. A. (1979). "Spectrum of Relict Gravitational Radiation and The Early State of the Universe" (PDF). JETP Lett. 30, 682 (Pisma Zh. Eksp. Teor. Fiz. 30, 719).
- Linde, Andrei (November 1994). "The Self-Reproducing Inflationary Universe" (PDF). Scientific American: page 51.
- Linde, A.D. (August 1986). "Eternally Existing Self-Reproducing Chaotic Inflationary Universe" (PDF). Physics Letters B 175 (4): 395–400. Bibcode:1986PhLB..175..395L. doi:10.1016/0370-2693(86)90611-8.
- Linde, A. (1986). "Eternal Chaotic Inflation". Mod. Phys. Lett. A1 (2): 81. Bibcode:1986MPLA....1...81L. doi:10.1142/S0217732386000129.
- Stenger, Victor J. "Is the Universe fine-tuned for us?" (PDF).
- Aguirre, Anthony and Gratton, Steven n (2003). "Inflation without a beginning: A null boundary proposal". Phys. Rev. D 67 (8). arXiv:gr-qc/0301042. Bibcode:2003PhRvD..67h3515A. doi:10.1103/PhysRevD.67.083515.
- Aguirre, Anthony, and Gratton, Steven (2002). "Steady-State Eternal Inflation". Phys. Rev. D 65 (8). arXiv:astro-ph/0111191. Bibcode:2002PhRvD..65h3507A. doi:10.1103/PhysRevD.65.083507.
- Gribbin, John. "Inflation for Beginners".
- See Linde (1990) and Mukhanov (2005).
- Mukhanov, Viatcheslav F.; Chibisov, G. V. (1981). "Quantum fluctuation and "nonsingular" universe". JETP Lett. 33: 532–5. Bibcode:1981JETPL..33..532M.
- Mukhanov, Viatcheslav F. (1982). "The vacuum energy and large scale structure of the universe". Sov. Phys. JETP 56: 258–65.
- See Guth (1997) for a popular description of the workshop, or The Very Early Universe, ISBN 0521316774 eds Hawking, Gibbon & Siklos for a more detailed report
- Hawking, S.W. (1982). "The development of irregularities in a single bubble inflationary universe". Phys. Lett. B115: 295. Bibcode:1982PhLB..115..295H. doi:10.1016/0370-2693(82)90373-2.
- Starobinsky, Alexei A. (1982). "Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations". Phys. Lett. B117: 175–8. Bibcode:1982PhLB..117..175S. doi:10.1016/0370-2693(82)90541-X.
- Guth, A.H. (1982). "Fluctuations in the new inflationary universe". Phys. Rev. Lett. 49 (15): 1110–3. Bibcode:1982PhRvL..49.1110G. doi:10.1103/PhysRevLett.49.1110.
- Bardeen, James M. (1983). "Spontaneous creation Of almost scale-free density perturbations in an inflationary universe". Phys. Rev. D28: 679. Bibcode:1983PhRvD..28..679B. doi:10.1103/PhysRevD.28.679.
- Weinberg, Steven (2006-11-05). "Beyond Belief: Science, Reason, Religion & Survival, Session 1". Salk Institute: The Science Network. 13:00–14:10. Retrieved 2012-08-27.
Just in recent years, through developments in the theory of the very early universe — in particular, the theory of chaotic inflation due to Andrei Linde — we now have a picture which is, I would say, plausible but not yet well established, that our Big Bang... is just one episode in a much larger multiverse, in which Big Bangs — or maybe I should say, not-so-Big Bangs are popping off all the time.
- 'Multiverse' theory suggested by microwave background BBC News, 3 August 2011 about testing eternal inflation.