A hypersurface in a Riemannian manifold is called minimal if its mean curvature vanishes identically.  Minimal surfaces have fascinated mathematicians since the time of Euler, and tremendous progress has been made in understanding the structure of the space of embedded minimal surfaces in various ambient manifolds in the last 50 years.  After surveying the subject, I will discuss recent developments concerning the uniqueness of the critical catenoid and the construction of infinitely many embedded minimal surfaces by ”gluing”.  The latter project is joint with N. Kapouleas.