In this paper, we consider the application of compressed sensing (aka compressive sampling) to speech and audio signals. We discuss the design considerations and issues that must be addressed in doing so, and we apply compressed sensing as a pre-processor to sparse decompositions of real speech and audio signals using dictionaries composed of windowed complex sinusoids. Our results demonstrate that the principles of compressed sensing can be applied to sparse decompositions of speech and audio signals and that it offers a significant reduction of the computational complexity, but also that such signals may pose a challenge due to their non-stationary and complex nature with varying levels of sparsity.