The emergence of compressive sensing and the associated ℓ1 recovery algorithms and theory have generated considerable excitement and interest in their applications. This chapter will examine recent developments and a complementary set of tools based on a Bayesian framework to address the general problem of sparse signal recovery and the challenges associated with it. Bayesian methods offer superior performance compared to convex optimization-based methods and are parameter tuning-free. They also have the flexibility necessary to deal with a diverse range of measurement modalities and structured sparsity in signals than hitherto possible. Parsimonious signal representation using overcomplete dictionaries for compression, estimation of sparse communication channels with large delay spread as in underwater acoustics, low-dimensional representation of MIMO wireless channels, brain imaging techniques, such as MEG and EEG, are a few examples. We provide a mathematically rigorous and in-depth overview of this fascinating area within sparse signal recovery. We highlight the generality and flexibility of Bayesian approaches and show how it greatly facilitates their deployment in communications-related applications, even though they generally lead to nonconvex optimization problems. Further, we show that, by reinterpreting the Bayesian cost function as a technique to perform covariance matching, one can develop new, ultrafast Bayesian algorithms for sparse signal recovery. As an example application, we discuss the utility of these algorithms in the context of 5G communications with several case studies including wideband time-varying channel estimation and low-resolution analog-to-digital conversion-based signal recovery.@en