Sparse approximation is a well-established theory, with a profound impact on the fields of signal and image processing. In this talk, we start by presenting this model and its features, and then turn to describe two special cases from it: 1) the convolutional sparse coding (CSC) and 2) its multi-layered version (ML-CSC). Amazingly, as we will carefully show, ML-CSC provides a solid theoretical foundation to deep-learning architectures.