A classic problem in mathematics and computer science is that of manifold learning. Given an unknown compact manifold embedded in Euclidean space and a known uniform sample of points lying on or near that manifold, how much of the manifold's structure can be inferred from the points alone? We will survey some fantastic work of Niyogi, Smale and Weinberger which provides explicit bounds on the number of points required to reconstruct the manifold in question up to homotopy type with high confidence. Next, we imagine the situation where an unknown function between two such manifolds must be inferred from point samples lying near the domain and range manifolds as well as the ability to evaluate the function at the domain- sample. We will see that not only can such functions be reconstructed up to homotopy from finite point samples with high confidence, but also that the entire process of reconstruction is robust to certain models of sampling and evaluation noise.