Review of recent works devoted to the variation of the fine-structure constant α=e²/ħc, strong interaction and fundamental masses (Higgs vacuum) is presented. Theories unifying gravity with other interactions suggest temporal and spatial variation of the fundamental “constants” in expanding Universe. The spatial variation can explain fine tuning of the fundamental constants which allows humans (and any life) to appear. We appeared in the area of the Universe where the values of the fundamental constants are consistent with our existence. There are some hints for the variation in quasar absorption spectra and Big-Bang nucleosynthesis data. However, a majority of publications report only limits on possible variation. A stringent limit on the variation of the strong interaction and fundamental masses follows from the Oklo natural nuclear reactor data. A very promising method to search for the variation consists in comparison of different atomic clocks. Huge enhancement of the relative variation effects happens in transitions between close atomic and molecular energy levels. We suggest several new cases where the levels are very narrow. A new idea is to build a “nuclear” clock based on UV (7.6 eV) transition in thorium nucleus. This may allow to improve sensitivity to the variation by several orders of magnitude. Large enhancement of the variation effects is also possible in cold atomic and molecular collisions near Feshbach resonance. How changing physical constants and violation of local position invariance may occur? The fundamental constants may depend on scalar fields which very naturally appear in modern cosmological models. Space-time evolution of these fields in expanding Universe may lead to the variation of the fundamental constants. Massive bodies (galaxies, stars, planets) can also affect physical constants. They have large scalar charge S proportional to number of particles which produces a Coulomb-like scalar field U=S/r. This leads to a variation of the fundamental constants proportional to the gravitational potential, e.g., δα/α=k_αδ(GM/rc²). We compare different manifestations of this effect. The strongest limits are obtained from the measurements of dependence of atomic frequencies on the distance from Sun (the distance varies due to the ellipticity of the Earth's orbit).