A great deal of theoretic and algorithmic research has revolved around sparsity view of signals over the last decade to characterize new, sub-Nyquist sampling limits as well as tractable algorithms for signal recovery from dimensionality reduced measurements. Despite the promising advances made, real-life applications require more realistic signal models that can capture the underlying, application-dependent order of sparse coefficients, better sampling matrices with information preserving properties that can be implemented in practical systems, and ever faster algorithms with provable recovery guarantees for real-time operation.