According to eternal inflation, the inflationary phase of the universe's expansion lasts forever throughout most of the universe. Because the regions expand exponentially rapidly, most of the volume of the universe at any given time is inflating. Eternal inflation, therefore, produces a hypothetically infinite multiverse, in which only an insignificant fractal volume ends inflation.
Alan Guth's 2007 paper, "Eternal inflation and its implications", detailed what was known about the subject at the time, and demonstrated that eternal inflation was still considered the likely outcome of inflation, more than 20 years after it was first introduced by Steinhardt.
Development of the Theory
Inflation, or the inflationary universe theory, was originally developed as a way to overcome the few remaining problems with what was otherwise considered a successful theory of cosmology, the Big Bang model.
In 1979, Alan Guth introduced the inflationary model of the universe to explain why the universe is flat and homogeneous (which refers to the smooth distribution of matter and radiation on a large scale).  The basic idea was that the universe underwent a period of rapidly accelerating expansion a few instants after the Big Bang. He offered a mechanism for causing the inflation to begin: false vacuum energy. Guth coined the term "inflation," and was the first to discuss the theory with other scientists worldwide.
Guth's original formulation was problematic, as there was no consistent way to bring an end to the inflationary epoch and end up with the hot, isotropic, homogeneous universe observed today. Although the false vacuum could decay into empty "bubbles" of "true vacuum" that expanded at the speed of light, the empty bubbles could not coalesce to reheat the universe, because they could not keep up with the remaining inflating universe.
In 1982, this "graceful exit problem" was solved independently by Andrei Linde and by Andreas Albrecht and Paul J. Steinhardt. who showed how to end inflation without making empty bubbles and, instead, end up with a hot expanding universe. The basic idea was to have a continuous "slow-roll" or slow evolution from false vacuum to true without making any bubbles. The improved model was called "new inflation."
In 1983, Paul Steinhardt was the first to show that this "new inflation" does not have to end everywhere.  Instead, it might only end in a finite patch or a hot bubble full of matter and radiation, and that inflation continues in most of the universe while producing hot bubble after hot bubble along the way. Alexander Vilenkin showed that when quantum effects are properly included, this is actually generic to all new inflation models.
Using ideas introduced by Steinhardt and Vilenkin, Andrei Linde published an alternative model of inflation in 1986, entitled "Eternally Existing Self-Reproducing Chaotic Inflationary Universe" which used these ideas to provide a detailed description of what has become known as the Chaotic Inflation theory or eternal inflation.,
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.
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.
- Gibbons, Gary W.; Hawking, Stephen W.; Siklos, S.T.C., eds. (1983). "Natural Inflation," in "The Very Early Universe. Cambridge University Press. pp. 251–66. ISBN 0-521-31677-4.
- Vilenkin, Alexander (1983). "Birth of Inflationary Universes". Physical Review D. 27 (12): 2848–2855. Bibcode:1983PhRvD..27.2848V. doi:10.1103/PhysRevD.27.2848.
- Guth, Alan; Eternal inflation and its implications arXiv:hep-th/0702178
- Guth, Alan H. "Inflationary universe: A possible solution to the horizon and flatness problems". Phys. Rev. D. 23 (2): 347–356. doi:10.1103/PhysRevD.23.347.
- Albrecht, A.; Steinhardt, P. J. (1982). "Cosmology For Grand Unified Theories With Radiatively Induced Symmetry Breaking". Phys. Rev. Lett. 48 (17): 1220–1223. Bibcode:1982PhRvL..48.1220A. doi:10.1103/PhysRevLett.48.1220.
- 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". Modern Physics Letters A. 1 (2): 81–85. Bibcode:1986MPLA....1...81L. doi:10.1142/S0217732386000129.
- Stenger, Victor J. "Is the Universe fine-tuned for us?" (PDF).
- Aguirre, Anthony; Gratton, Steven n (2003). "Inflation without a beginning: A null boundary proposal". Physical Review D. 67 (8): 083515. arXiv:. Bibcode:2003PhRvD..67h3515A. doi:10.1103/PhysRevD.67.083515.
- Aguirre, Anthony; Gratton, Steven (2002). "Steady-State Eternal Inflation". Physical Review D. 65 (8): 083507. arXiv:. Bibcode:2002PhRvD..65h3507A. doi:10.1103/PhysRevD.65.083507.
- Gribbin, John. "Inflation for Beginners". Archived from the original on September 6, 2014.
- See Linde (1990) and Mukhanov (2005).
- Mukhanov, Viatcheslav F.; Chibisov, G. V. (1981). "Quantum fluctuation and "nonsingular" universe". JETP Letters. 33: 532–5. Bibcode:1981JETPL..33..532M.
- Mukhanov, Viatcheslav F. (1982). "The vacuum energy and large scale structure of the universe". Soviet Physics 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". Physics Letters B. 115 (4): 295–297. 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". Physics Letters B. 117 (3–4): 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". Physical Review D. 28 (4): 679–693. 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.
- Our Universe May Exist in a Multiverse, Cosmic Inflation Discovery Suggests
- 'Multiverse' theory suggested by microwave background BBC News, 3 August 2011 about testing eternal inflation.