Reconfigurable intelligent surface (RIS)-assisted millimeter wave (mmWave) communication is a promising technology for enlarging the coverage area of millimeter wave systems. Unfortunately, realizing the full potential of these systems requires addressing numerous challenges in channel estimation. In this paper, channel estimation for RIS-assisted mmWave communications is considered. Under the assumption that the array manifolds of the base station antennas and the RIS reflecting elements are given by uniform arrays, an efficient two-stage channel estimation method based on array signal processing techniques is proposed. In the proposed algorithm, the direct and indirect channels are jointly estimated by space-time processing that exploits the sparsity in RIS-assisted mmWave channels and the features associated with uniform arrays. Then, several practical issues, including detection of the number of channel paths, imperfect RIS hardware, and complexity, are addressed. Extensions to the cases of uniform planar array-based RIS, wideband communication, and multiple users are also discussed. Numerical results validate the effectiveness of the proposed method.