Iterative shrinkage/thresholding (IST) algorithms are an important component of the computational toolbox used to address linear inverse problems under sparseness-inducing regularization. Examples of this class of problems include (among many others) wavelet-based and total-variation-based image restoration, compressive sensing, and coded aperture imaging. IST algorithms are typically used to address unconstrained minimization problems where the objective function includes a quadratic data term (for linear-Gaussian observations) and a sparseness-inducing regularizer (typically a 1-norm).  Via the recently proposed Bregman iterative approach, IST algorithms also play a central role in dealing with the related constrained optimization formulations. The purpose of this talk is threefold: (a) To review the several (at least 4) different perspectives from which IST algorithms can be built/derived, as well as several convergence results. (b) To present recent advances, namely several fast variants of IST which are the current state-of-the-art in solving L2-L1 (and related) problems. (c) To describe recent extensions, such as those designed for non-Gaussian noise and for multiple regularizers.