Parameterized matrices or linear systems arise in many applications. Often we need to solve such linear systems for a large number of parameters, for example, for a predetermined sweep of parameter values or while sampling from parameter space. Rather than computing a new preconditioner for each parameter or using a single preconditioner for all parameters, we can often update a high quality preconditioner computed for one system, at low cost, to be just as good for a subsequent system. The key idea in our approach is to consider updating a preconditioner as computing an approximate map from a new matrix to a previous matrix. We discuss a few approaches to updating preconditioners, their analysis, and several applications.