While there is strong motivation for using Gaussian Processes (GPs) due to their excellent performance in regression and classiﬁcation problems, their computational complexity makes them impractical when the size of the training set exceeds a few thousand cases. This has motivated the recent proliferation of a number of cost-eﬀective approximations to GPs, both for classiﬁcation and for regression. In this paper we analyze one popular approximation to GPs for regression: the reduced rank approximation. While generally GPs are equivalent to inﬁnite linear models, we show that Reduced Rank Gaussian Processes (RRGPs) are equivalent to ﬁnite sparse linear models. We also introduce the concept of degenerate GPs and show that they correspond to inappropriate priors. We show how to modify the RRGP to prevent it from being degenerate at test time. Training RRGPs consists both in learning the covariance function hyperparameters and the support set. We propose a method for learning hyperparameters for a given support set. We also review the Sparse Greedy GP (SGGP) approximation (Smola and Bartlett, 2001), which is a way of learning the support set for given hyperparameters based on approximating the posterior. We propose an alternative method to the SGGP that has better generalization capabilities. Finally we make experiments to compare the diﬀerent ways of training a RRGP. We provide some Matlab code for learning RRGPs.