Title: Continual Learning with Soft-Masking of Parameter-Level Gradient Flow. Abstract: Existing research on task incremental learning in continual learning has primarily focused on preventing catastrophic forgetting (CF). Several techniques have achieved learning with no CF. However, they attain it by letting each task monopolize a sub-network in a shared network, which seriously limits knowledge transfer (KT) and causes over-consumption of the network capacity, i.e., as more tasks are learned, the performance deteriorates. The goal of this paper is threefold: (1) overcoming CF, (2) encouraging KT, and (3) tackling the capacity problem. A novel and simple technique (called SPG) is proposed that soft-masks (partially blocks) parameter updating in training based on the importance of each parameter to old tasks. Each task still uses the full network, i.e., no monopoly of any part of the network by any task, which enables maximum KT and reduction of capacity usage. Extensive experiments demonstrate the effectiveness of SPG in achieving all three objectives. More notably, it attains significant transfer of knowledge not only among similar tasks (with shared knowledge) but also among dissimilar tasks (with little shared knowledge) while preventing CF.