This paper describes the experimental framework and results of the ICDAR 2021 Competition on On-Line Signature Verification (SVC 2021). The goal of SVC 2021 is to evaluate the limits of on-line signature verification systems on popular scenarios (office/mobile) and writing inputs (stylus/finger) through large-scale public databases. Three different tasks are considered in the competition, simulating realistic scenarios as both random and skilled forgeries are simultaneously considered on each task. The results obtained in SVC 2021 prove the high potential of deep learning methods. In particular, the best on-line signature verification system of SVC 2021 obtained Equal Error Rate (EER) values of 3.33% (Task 1), 7.41% (Task 2), and 6.04% (Task 3). SVC 2021 will be established as an on-going competition, where researchers can easily benchmark their systems against the state of the art in an open common platform using large-scale public databases such as DeepSignDB and SVC2021_EvalDB, and standard experimental protocols.