This review is devoted to critical phenomena such as the threshold of explosive instability and kinetic transitions of the ``medium populating'' type in nonequilibrium systems with breeding, decay, and diffusion. A detailed analysis is made of the situation where breeding is localized within particular spatial regions (breeding centers) which arise randomly in the medium at arbitrary times and have finite lifetimes. The analogy with problems in percolation theory and second- order equilibrium phase transitions is discussed. The effect of fluctuations in external fields on competition processes in media with diffusion is examined. Diffusion in a medium with randomly distributed traps is investigated and particular attention is devoted to the contribution of statistically rare spatial configurations.