Existing methods for robust multigroup multicast beamforming obtain feasible points using semidefinite relaxation (SDR) and Gaussian randomization, and have high computational complexity. In this letter, we consider the robust multigroup multicast beamforming design to minimize the sum power (SP) or per-antenna power (PAP) under the signal-to-interference-plus-noise ratio (SINR) constraints and to maximize the worst-case SINR under the SP constraint or PAP constraints, respectively. The resulting optimization problems are challenging non-convex problems with infinitely many constraints. For each problem, using the majorization-minimization (MM) approach, we propose an iterative algorithm to obtain a feasible solution which is shown to be a stationary point under certain conditions. We also show that the proposed algorithms have much lower computational complexity than existing SDR-based algorithms.