In-band Full-Duplex (FD) Multiple-Input Multiple-Output (MIMO) systems offer a significant opportunity for Integrated Sensing and Communications (ISAC) due to their capability to realize simultaneous signal transmissions and receptions. This feature has been recently exploited to devise spectrum-efficient simultaneous information transmission and monostatic sensing operations, a line of research typically referred to as MIMO FD-ISAC. In this article, capitalizing on a recent FD MIMO architecture with reduced complexity analog cancellation, we present an FD-enabled framework for simultaneous communications and sensing using data signals. In contrast to communications applications, the framework's goal is not to mitigate self interference, since it includes reflections of the downlink data transmissions from targets in the FD node's vicinity, but to optimize the system parameters for the intended dual functionality. The unique characteristics and challenges of a generic MIMO FD-ISAC system are discussed along with a broad overview of state-of-the-art special cases, including numerical investigations. Several directions for future work on FD-enabled ISAC relevant to signal processing communities are also provided.