Recent Progress on High Tc Superconductivity and Related Problems

High Tc superconductivity in cuprates has engaged us for over 25 years. An apparent quantum complexityn arising from strong electron correlations and Mott physics, in the form of new experimental results, continues to challenge theorists. Intense efforts from theory has brought in a good and deep understanding of the optimal doped region of the phase diagram, particularly the superconducting phase.  However, anomalous normal state in the optimal and under doped regime has defied simple theoretical understanding, partly because of nearly degenerate competing phases, differing structures and quantum chemistry among members of the cuprate family. But a standard model for the description of the winning superconducting phase is definitely there. There are also signals for some simplicity and new physic behind the complexity.