We address multiview stereo (MVS), an important 3D vision task that reconstructs a 3D model such as a dense point cloud from multiple calibrated images. We propose CER-MVS (Cascaded Epipolar RAFT Multiview Stereo), a new approach based on the RAFT (Recurrent All-Pairs Field Transforms) architecture developed for optical flow. CER-MVS introduces five new changes to RAFT: epipolar cost volumes, cost volume cascading, multiview fusion of cost volumes, dynamic supervision, and multiresolution fusion of depth maps. CER-MVS is significantly different from prior work in multiview stereo. Unlike prior work, which operates by updating a 3D cost volume, CER-MVS operates by updating a disparity field. Furthermore, we propose an adaptive thresholding method to balance the completeness and accuracy of the reconstructed point clouds. Experiments show that our approach achieves state-of-the-art performance on the DTU and Tanks-and-Temples benchmarks (both intermediate and advanced set). Code is available at https://github.com/princeton-vl/CER-MVS.