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Biological tissues are a complex mixture of specialized cell types which display heterogeneity at the level of gene expression and in some cases at the level of the genome. Single cell studies thus have the potential to reveal important biological insights that are hidden in bulk measurements. However, realizing this potential requires overcoming technical challenges unique to single cell measurements. Single cell sequencing requires amplification, which introduces errors that must be minimized. Moreover, dissecting a heterogeneous system requires methods capable of examining large numbers of cells in a timely and cost-effective manner. We present a set of methods for single cell whole genome and transcriptome amplification that provide high accuracy and sensitivity and readily scale to allow analysis of thousands of cells. Using these methods, we present applications to cancer and developmental biology and demonstrate that biological insights can be revealed by measuring noise within a homogenous population.