This paper presents a unified convergence analysis of block successive minimization methods for nonsmooth optimization. The authors study an inexact block coordinate descent (BCD) approach that updates variable blocks by successively minimizing approximations of the objective function, which are either locally tight upper bounds or strictly convex local approximations. The analysis focuses on characterizing the convergence properties of a wide class of such methods, including the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, and the alternating proximal minimization algorithm. The paper introduces the Successive Upper-bound Minimization (SUM) algorithm, which is a simple approach where all variables are grouped into a single block. The SUM algorithm is shown to converge to a stationary point under certain mild assumptions on the approximation function. The authors then extend this analysis to the Block Successive Upper-bound Minimization (BSUM) algorithm, which updates one block at a time. The BSUM algorithm is shown to converge to a coordinatewise minimum of the objective function under quasi-convexity assumptions, and to converge to a stationary point under compactness assumptions on the level sets. The paper also considers the Maximum Improvement Successive Upper-bound Minimization (MISUM) algorithm, which updates the block that provides the maximum improvement at each step. The MISUM algorithm is shown to converge to a coordinatewise minimum of the objective function without requiring the uniqueness of the minimizer for each subproblem. In addition, the paper extends the analysis to the Block Successive Convex Approximation (BSCA) algorithm, which uses convex approximations of the objective function. The BSCA algorithm is shown to converge to a stationary point under certain smoothness and convexity assumptions. The paper also considers the overlapping essentially cyclic rule, a general block scheduling rule that allows for overlapping updates of variable blocks. The convergence results for the BSUM and BSCA algorithms are extended to this rule. The paper concludes by discussing applications of these algorithms, including linear transceiver design in cellular networks. The algorithms are shown to be effective in optimizing the sum of users' rates in a wireless network with transmit power constraints.This paper presents a unified convergence analysis of block successive minimization methods for nonsmooth optimization. The authors study an inexact block coordinate descent (BCD) approach that updates variable blocks by successively minimizing approximations of the objective function, which are either locally tight upper bounds or strictly convex local approximations. The analysis focuses on characterizing the convergence properties of a wide class of such methods, including the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, and the alternating proximal minimization algorithm. The paper introduces the Successive Upper-bound Minimization (SUM) algorithm, which is a simple approach where all variables are grouped into a single block. The SUM algorithm is shown to converge to a stationary point under certain mild assumptions on the approximation function. The authors then extend this analysis to the Block Successive Upper-bound Minimization (BSUM) algorithm, which updates one block at a time. The BSUM algorithm is shown to converge to a coordinatewise minimum of the objective function under quasi-convexity assumptions, and to converge to a stationary point under compactness assumptions on the level sets. The paper also considers the Maximum Improvement Successive Upper-bound Minimization (MISUM) algorithm, which updates the block that provides the maximum improvement at each step. The MISUM algorithm is shown to converge to a coordinatewise minimum of the objective function without requiring the uniqueness of the minimizer for each subproblem. In addition, the paper extends the analysis to the Block Successive Convex Approximation (BSCA) algorithm, which uses convex approximations of the objective function. The BSCA algorithm is shown to converge to a stationary point under certain smoothness and convexity assumptions. The paper also considers the overlapping essentially cyclic rule, a general block scheduling rule that allows for overlapping updates of variable blocks. The convergence results for the BSUM and BSCA algorithms are extended to this rule. The paper concludes by discussing applications of these algorithms, including linear transceiver design in cellular networks. The algorithms are shown to be effective in optimizing the sum of users' rates in a wireless network with transmit power constraints.