In many theories beyond the Standard Model the quantities that we call "fundamental constants" become space-time dependent, leading to corresponding variation of atomic and molecular spectra and clock frequencies. The extraordinary improvement of the atomic clock precision in the past fifteen years enabled testing the constancy of the fundamental constant at a very high level of precision. Herein, searches for the variation of fundamental constants with clocks are discussed, focusing on recent key results and future proposals, including highly charged ion, molecular, and nuclear clocks. The relevance of the recent searches for oscillatory and transient variation of fundamental constants to the major unexplained phenomena of our Universe, the nature of dark matter, is discussed.