For quasilinear partial differential and integrodifferential equations and inequalities containing nonlinearities of the Kardar–-Parisi–-Zhang type, various (old and recent) results on qualitative properties of solutions (such as the stabilization of solutions, blow-up phenomena, long-time decay of solutions, and others) are presented. Descriptive examples demonstrating the Bitsadze approach (the technique of monotone maps) applied in this research area are provided.