Many fundamental scientific and engineering problems of recent interest arising in signal recovery, image processing, compressive sensing, machine learning and other fields can be formulated as well structured optimizationproblems, but which are typically very large scale, and often nonsmooth. This leads to challenging difficulties for their solutions, precluding the use of second-order algorithms such as interior point methods. Elementary first order methods then often remain our best alternative to tackle such problems. This talk surveys classical as well as recent results on the design and analysis of gradient-based algorithms for some generic optimization models arising in a wide variety of applications, highlighting the ways in which problem structures can be beneficially exploited to devise simple and efficient algorithms.