In light of the finite nature of the wireless spectrum and the increasing demand for spectrum use arising from recent technological breakthroughs in wireless communication, the problem of interference continues to persist. Despite recent advancements in resolving interference issues, interference still presents a difficult challenge to effective usage of the spectrum. This is partly due to the rise in the use of license-free and managed shared bands as well as other opportunistic spectrum access solutions. As a result of this, the need for efficient spectrum usage schemes that are robust against interference has never been more important. In the past, most solutions to interference have addressed the problem by using avoidance techniques as well as mitigation approaches based on expert systems. More recently, researchers have successfully explored artificial intelligence/machine learning enabled physical layer techniques, especially deep learning which reduces or compensates for the interfering signal instead of simply avoiding it. In this paper, we address the knowledge gap in literature with respect to the state-of-the-art in deep learning-based interference suppression. Specifically, we review a wide range of techniques that have used deep learning to suppress interference by learning interference characteristics directly from data, rather than relying on expert systems. We provide a thorough technical discussion of the prominent deep learning algorithms that have been proposed in the literature and provide comparison and guidelines regarding their successful implementation in this application. In addition, we highlight challenges and potential future research directions for the successful adoption of deep learning in this critical field.