We consider the problem of how to design and implement communication-efficient versions of parallel support vector machines, a widely used classifier in statistical machine learning, for distributed memory clusters and supercomputers. The main computational bottleneck is the training phase, in which a statistical model is built from an input data set. Prior to our study, the parallel isoefficiency of a state-of-the-art implementation scaled as W = Omega(P^3), where W is the problem size and P the number of processors; this scaling is worse than even a one-dimensional block row dense matrix vector multiplication, which has W = Omega(P^2). This study considers a series of algorithmic refinements, leading ultimately to a Communication-Avoiding SVM (CASVM) method that improves the isoefficiency to nearly W = Omega(P). We evaluate these methods on 96 to 1536 processors, and show average speedups of 3 - 16x (7x on average) over Dis-SMO, and a 95% weak-scaling efficiency on six realworld datasets, with only modest losses in overall classification accuracy. The source code can be downloaded at [1].