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. Another possible effect is dependence of the fundamental constant on the gravitational potential which leads t the violation of local position invariance.
I present a review of recent works devoted to the variation of the fine structure constant alpha, strong interaction and fundamental masses (Higgs vacuum). There are some hints for the variation in quasar absorption spectra, Big Bang nucleosynthesis, and Oklo natural nuclear reactor data.
A very promising method to search for the variation consists in comparison of different atomic clocks. Hugs enhancement of the relative variation effects happens in transitions between close atomic, molecular and nuclear energy levels. We suggest several new cases where the levels are very narrow. 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? Light scalar fields very naturally appear in modern cosmological models, affecting parameters of the Standard Model (e.g. alpha). Cosmological variations of these scalar fields should occur because of drastic changes of matter composition in Universe: the latest such event is rather recent (about 5 billion years ago), from matter to dark energy domination. Massive bodies (stars or galaxies) 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. We compared 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).