Among all surfaces within a given boundary configuration, surfaces that minimize area have vanishing mean curvature. In this expository talk we will introduce minimal surfaces as zero mean curvature surfaces. We will see that they correspond to a pair of meromorphic and holomorphic functions and use this to answer the geometric problem of finding a minimal surface that passes through a given curve with prescribed tangent planes. As a consequence we will see that given any regular real analytic curve, there is a minimal surface containing it as a geodesic.