This review presents an overview of the recent developments and achievements in the field of nonlinear microwave physics and applications of high temperature superconducting (HTS) thin films. It touches upon such issues as power-dependent surface impedance Z(s), harmonic generation and intermodulation distortion. Various possible sources of the nonlinearity including grain boundaries, patterning, impurity doping, oxygen content and microwave heating are analysed and surveyed. Correlative studies of microstructural, microwave and other electromagnetic properties, as well as techniques of artificial modification of HTS films with the aim to improve their microwave performance, are reviewed. Possible perspectives for near future development of the nonlinear microwave science and application of HTS films are also discussed.