Currently, the increasing number of daily flights emphasizes the importance of air transportation. Furthermore, Air Traffic Management (ATM) enables air carriers to operate safely and efficiently through the multiple services provided. Advanced analytic solutions have demonstrated the potential to solve complex problems in several domains, and Deep Learning (DL) has attracted attention due to its impressive results and disruptive capabilities. The adoption of DL models in ATM solutions enables new cognitive services that have never been considered before. The main goal of this research is to present a comprehensive review of state-of-the-art Deep Learning (DL) solutions for Air Traffic Management (ATM). This review focuses on describing applications, identifying opportunities, and highlighting open challenges to foster the evolution of ATM systems. To accomplish this, we discuss the fundamental topics of DL and ATM and categorize the contributions based on different approaches. First, works are grouped based on the DL approach adopted. Then, future directions are identified based on the ATM solution area. Finally, open challenges are listed for both DL applications and ATM solutions. This article aims to support the community by identifying research problems to be faced in the future.