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Many problems in signal and image processing, machine learning, and estimation require optimization of convex cost functions. For convex cost functions with Lipschitz continuous gradients, Nesterov's fast gradient method decreases the cost function at least as fast as the square of the number of iterations, a rate order that is optimal. This talk describes a new first-order optimization method called the optimized gradient method (OGM) that converges twice as fast as Nesterov's famous method yet has a remarkably similar simple implementation. Interestingly, Drori recently showed that OGM has optimal complexity among first-order methods. I will discuss other recent extensions and show examples in machine learning and X-ray computed tomography (CT). This work is joint with Donghwan Kim.