Variation of fundamental constants in space and time: theory and observations

Review of recent works devoted to the temporal and spatial variation of the fundamental constants and dependence of the fundamental constants on the gravitational potential (violation of local position invariance) is presented. We discuss the variation of the fine structure constant $\alpha=e^2/\hbar c$, strong interaction and fundamental masses (Higgs vacuum), e.g. the electron-to-proton mass ratio $\mu=m_e/M_p$ or $X_e=m_e/\Lambda_{QCD}$ and $X_q=m_q/\Lambda_{QCD}$. We also present new results from Big Bang nucleosynthesis and Oklo natural nuclear reactor data and propose new measurements of enhanced effects in atoms, nuclei and molecules, both in quasar and laboratory spectra.