  Isotropic models in loop quantum cosmology allow explicit calculations,
thanks largely to a completely known volume spectrum, which is exploited in
order to write down the evolution equation in a discrete internal time. Because
of genuinely quantum geometrical effects the classical singularity is absent in
those models in the sense that the evolution does not break down there,
contrary to the classical situation where space-time is inextendible. This
effect is generic and does not depend on matter violating energy conditions,
but it does depend on the factor ordering of the Hamiltonian constraint.
Furthermore, it is shown that loop quantum cosmology reproduces standard
quantum cosmology and hence (e.g., via WKB approximation) to classical behavior
in the large volume regime where the discreteness of space is insignificant.
Finally, an explicit solution to the Euclidean vacuum constraint is discussed
which is the unique solution with semiclassical behavior representing quantum
Euclidean space.
