This paper presents the third data release of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP), a wide-field multi-band imaging survey with the Subaru 8.2 m telescope. HSC-SSP has three survey layers (Wide, Deep, and UltraDeep) with different area coverages and depths, designed to address a wide array of astrophysical questions. This third release from HSC-SSP includes data from 278 nights of observing time and covers about 670 deg2 in all five broad-band filters (grizy) at the full depth (∼26 mag at 5σ depending on filter) in the Wide layer. If we include partially observed areas, the release covers 1470 deg2. The Deep and UltraDeep layers have ∼ 80% of the originally planned integration times, and are considered done, as we have slightly changed the observing strategy in order to compensate for various time losses. There are a number of updates in the image processing pipeline. Of particular importance is the change in the sky subtraction algorithm; we subtract the sky on small scales before the detection and measurement stages, which has significantly reduced the number of false detections. Thanks to this and other updates, the overall quality of the processed data has improved since the previous release. However, there are limitations in the data (for example, the pipeline is not optimized for crowded fields), and we encourage the user to check the quality assurance plots as well as a list of known issues before exploiting the data.