This paper summarizes in depth the state of the art of aerial swarms, covering both classical and new reinforcement-learning-based approaches for their management. Then, it proposes a hybrid AI system, integrating deep reinforcement learning in a multi-agent centralized swarm architecture. The proposed system is tailored to perform surveillance of a specific area, searching and tracking ground targets, for security and law enforcement applications. The swarm is governed by a central swarm controller responsible for distributing different search and tracking tasks among the cooperating UAVs. Each UAV agent is then controlled by a collection of cooperative sub-agents, whose behaviors have been trained using different deep reinforcement learning models, tailored for the different task types proposed by the swarm controller. More specifically, proximal policy optimization (PPO) algorithms were used to train the agents' behavior. In addition, several metrics to assess the performance of the swarm in this application were defined. The results obtained through simulation show that our system searches the operation area effectively, acquires the targets in a reasonable time, and is capable of tracking them continuously and consistently.