We propose a novel general piecewise surface regression model that allows for arbitrary functions to be used in each piece, and arbitrary boundary surfaces between pieces. We also give an EM-based algorithm for this model, EMPRR, that scales to high dimensions. We compare EMPRR's performance with those of model trees and functional trees, two regression tree learning methods, on synthetic piecewise data and benchmark data sets. Our results show that EMPRR outperforms the other two methods on the synthetic data sets and performs competitively on the benchmark data sets while generating accurate and compact models.