In this chapter, we explore various proximal algorithms for solving nonsmooth convex minimization problems. To this end, we begin by introducing the Legendre-Fenchel conjugate of convex functions and developing the corresponding subdifferential calculus. Next, we define the proximal mapping and investigate its key properties. Using the proximal mapping, we reformulate the underlying convex minimization problem as a fixed-point equation. This perspective naturally leads to a range of proximal algorithms, including the proximal point algorithm, the proximal gradient method and its acceleration, and the Douglas-Rachford splitting method. In particular, we provide a detailed study of the accelerated proximal gradient method, presenting state-of-the-art convergence results. These include the convergence of iterates and linear convergence under strong convexity. For a comprehensive survey on proximal algorithms, we refer the reader to Parikh and Boyd (Found Trends Optim 1(3):127–239, 2014).