The following necessary and sufficient criterion for an elementary theory $T$ to be model complete (cf. Model theory): for every two models $A$ and $B$ of $T$ such that $A$ is a substructure of $B$ (cf. Structure), it follows that $A$ is existentially closed in $B$.
Robinson test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Robinson_test&oldid=39937