Quantum cosmology is the attempt in theoretical physics to develop a quantum theory of the Universe. In this approach, it is all about to answer open questions of classical cosmology, particularly those related to the first phases of the Universe.
The classical cosmology is based on Albert Einstein's General Theory of Relativity (GTR). It describes the evolution of the Universe very well, as long as you do not approach the Big Bang. It is the gravitational singularity and the Planck time where relativity theory fails to provide what must be demanded of a final theory of space and time. Therefore, a theory is needed that integrates relativity theory and quantum theory. Such an approach is attempted for instance with the loop quantum gravity, another approach with the string theory.