Summary: Data recovery on a manifold is an important problem in many applications. Many such problems, e.g. compressive sensing, involve solving a system of linear equations knowing that the unknowns lie on a known manifold. The aim of this paper is to survey theoretical results and numerical algorithms about the recovery of signals lying on a manifold from linear measurements. Particularly, we focus on the case where signals lying on an algebraic variety. We first introduce the tools from algebraic geometry which plays an important role in studying the minimal measurement number and also show its applications. We finally introduce the numerical algorithms for solving it.