Conventional video compression (VC) methods are based on motion compensated transform coding, and the steps of motion estimation, mode and quantization parameter selection, and entropy coding are optimized individually due to the combinatorial nature of the end-to-end optimization problem. Learned VC allows end-to-end rate-distortion (R-D) optimized training of nonlinear transform, motion and entropy model simultaneously. Most works on learned VC consider end-to-end optimization of a sequential video codec based on R-D loss averaged over pairs of successive frames. It is well-known in conventional VC that hierarchical, bi-directional coding outperforms sequential compression because of its ability to use both past and future reference frames. This paper proposes a learned hierarchical bi-directional video codec (LHBDC) that combines the benefits of hierarchical motion-compensated prediction and end-to-end optimization. Experimental results show that we achieve the best R-D results that are reported for learned VC schemes to date in both PSNR and MS-SSIM. Compared to conventional video codecs, the R-D performance of our end-to-end optimized codec outperforms those of both x265 and SVT-HEVC encoders ("veryslow" preset) in PSNR and MS-SSIM as well as HM 16.23 reference software in MS-SSIM. We present ablation studies showing performance gains due to proposed novel tools such as learned masking, flow-field subsampling, and temporal flow vector prediction. The models and instructions to reproduce our results can be found in https://github.com/makinyilmaz/LHBDC/.