For parameterized systems, one standard problem is to determine the set of parameters which "best" fits given data.  Two examples of this will be summarized in this talk, both of which can be solved using homotopies.  The first is variational inference in which one searches in a parameterized family of probability distributions for a probability distribution that best fits the given data.  The second is synthesizing a linkage whose coupler curve best approximates the given data.  This talk is joint work with Emma Cobian, Fang Liu, and Daniele Schiavazzi (variational inference) and Aravind Baskar and Mark Plecnik (approximate synthesis).