A selective survey is given of convergence results for sequences of Pade approximants. Various approaches for dealing with the convergence problems due to "defects" are discussed. Attention is drawn to the close relationship between analyticity properties of a function and the "smoothness" of its Taylor series coefficients. A new theorem on the convergence of horizontal sequences of Pade approximants to functions in the Baker-Gammel-Wills conjecture function class is presented.