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In this paper, a comprehensive survey is given on several major systematic approaches in dealing with delay-aware control problems, namely the equivalentrate constraint approach, the Lyapunov stability drift approach, and the approximate Markov decision process approach using stochastic learning. These approaches essentially embrace most of the existing literature regarding delay-aware resource control in wireless systems. They have their relative pros and cons in terms of performance, complexity, and implementation issues. For each of the approaches, the problem setup, the general solution, and the design methodology are discussed. Applications of these approaches to delay-aware resource allocation are illustrated with examples in single-hop wireless networks. Furthermore, recent results regarding delay-aware multihop routing designs in general multihop networks are elaborated. Finally, the delay performances of various approaches are compared through simulations using an example of the uplink OFDMA systems.