A conformal field theory is a quantum field theory with extra symmetries (namely the conformal group). It can also be thought of as an invariant attached to a quantum field theory which captures its limiting behaviour at long distances or at short distances. Several quantum field theory can lead to the same conformal field theory at long (or short) distances, a phenomenon called "universality". Similarly, conformal field theories appear in the description of long range correlation functions of critical lattice models. In this talk I would like to illustrate the above ideas in the special case of two dimensions, discuss two related axiomatisations and outline some classification results.

In classical general relativity, singularities are the boundaries of the spacetime. At these events, spacetime curvature diverges and geodesic evolution breaks down. It has been long hoped that a quantum thoery of gravity will shed important insights on the resolution of singularities. In the last decade, progress in loop quantum cosmology has provided various robust results on singularity resolution in different cosmological models. Due to the underlying quantum gravitational effects, classical big bang is resolved and replaced by a bounce. In this talk we will provide an overview of some of main results in this direction, and discuss some new developments. These include numerical investigations of quantum spacetimes confirming bounce for highly quantum states and anisotropic models, and indications of generic resolution of strong curvature singularities.