Despite being a foundational concept of modern systems theory, there have been few studies on observability of non-linear stochastic systems under partial observations. In this paper, we introduce a definition of observability for stochastic non-linear dynamical systems which involves an explicit functional characterization. To justify its operational use, we establish that this definition implies filter stability under mild continuity conditions: an incorrectly initialized non-linear filter is said to be stable if the filter eventually corrects itself with the arrival of new measurement information. Numerous examples are presented and a detailed comparison with the literature is reported. We also establish implications for various criteria for filter stability under several notions of convergence such as weak convergence, total variation, and relative entropy. These findings are connected to robustness and approximations in partially observed stochastic control.