In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.

If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a₁, …, a_n) with parameters a₁, …, a_n from N is true in N if and only if it is true in M. If N is an elementary substructure of M, M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M.