Abstract: Conformal field theory in two dimensions (CFT), which is a successful theory of critical phenomena, is a promising holographic description of certain black hole physics. The celebrated Cardy formula, which gives the CFT entropy in the saddle point approximation, reproduces the Bekenstein-Hawking entropy of certain black holes. The Cardy formula is a direct consequence of modular invariance of the partition function. In this talk we review recent discoveries about modular invariant CFTs and their holographic implications.