Let $M$ be a manifold on which a notion of geodesic is defined. (For instance, $M$ could be a Riemannian manifold, $M$ could be a manifold with affine connection, or $M$ could be a Finsler space.)

Two distinct points, $P$ and $Q$ of $M$ are said to be conjugate points if there exist two or more distinct geodesic segments having $P$ and $Q$ as endpoints.

A simple example of conjugate points are the north and south poles of a sphere (endowed with the usual metric of constant curvature) -- every meridian is a geodesic segment having the poles as endpoints.