We present bunch compressor designs for the International Linear Collider (ILC) which achieve a reduction in RMS bunch length from 6 mm to 0.3 mm via multiple stages of compression, with stages of acceleration inserted between the stages of compression. The key advantage of multi-stage compression is that the maximum RMS energy spread is reduced to approximately 1 percent, compared to over 3 percent for a single-stage design. Analytic and simulation studies of the multi-stage bunch compressors are presented, along with performance comparisons to a single-stage system. Parameters for extending the systems to a larger total compression factor are discussed.