Recurrent neural networks trained with the backpropagation through time (BPTT) algorithm have led to astounding successes in various temporal tasks. However, BPTT introduces severe limitations, such as the requirement to propagate information backwards through time, the weight symmetry requirement, as well as update-locking in space and time. These problems become roadblocks for AI systems where online training capabilities are vital. Recently, researchers have developed biologically-inspired training algorithms, addressing a subset of those problems. In this work, we propose a novel learning algorithm called online spatio-temporal learning with target projection (OSTTP) that resolves all aforementioned issues of BPTT. In particular, OSTTP equips a network with the capability to simultaneously process and learn from new incoming data, alleviating the weight symmetry and update-locking problems. We evaluate OSTTP on two temporal tasks, showcasing competitive performance compared to BPTT. Moreover, we present a proof-of-concept implementation of OSTTP on a memristive neuromorphic hardware system, demonstrating its versatility and applicability to resource-constrained AI devices.