Starting with a carefully formulated Dirichlet process (DP) mixture model, we derive a generalized product partition model (GPPM) in which the partition process is predictor-dependent. The GPPM generalizes DP clustering to relax the exchangeability assumption through the incorporation of predictors, resulting in a generalized Pólya urn scheme. In addition, the GPPM can be used for formulating flexible semiparametric Bayes models for conditional distribution estimation, bypassing the need for expensive computation of large numbers of unknowns characterizing priors for dependent collections of random probability measures. A variety of special cases are considered, and an efficient Gibbs sampling algorithm is developed for posterior computation. The methods are illustrated using simulation examples and an epidemiologic application.