This chapter presents the main concepts and theoretical foundations related to sparsity-aware learning techniques. The concept of sparse modeling is introduced together with the LASSO and the ℓ0 and ℓ1 norm minimizing tasks. Conditions for uniqueness of the obtained solutions as well as for the equivalence of the ℓ0 and ℓ1 norm minimization are stated. The RIP condition and related bounds are discussed. Compressed sensing and the notion of stable embeddings are reviewed. The concept of sub-Nyquist sampling is presented and finally a case study concerning image de-nosing is reported.