Massive multiple-input multiple-output (MIMO) is a promising technique to enable orders of magnitude improvement in spectral and energy efficiency by utilizing a large number of antennas. Despite its theoretical gains, the implementation of large-scale antenna arrays faces many practical challenges in hardware cost, power consumption, and physical size. In this work, we study downlink massive MIMO systems in which the base stations (BSs) are equipped with dynamic metasurface antennas (DMAs). DMAs can realize low-cost, power-efficient, planar, and compact antenna arrays. We first formulate a mathematical model for DMA-based downlink massive MIMO systems. Then, we characterize the achievable sum-rate for the resulting systems and design an efficient alternating algorithm to dynamically configure DMA weights to maximize the achievable sum-rate. Our numerical results demonstrate that, in addition to their simplicity and low cost, properly configured DMAs can achieve downlink sum-rate performance which is comparable with the fundamental limits of multi-user MIMO systems.