Future wireless networks must support real-time, data-driven cyber-physical systems in which communication is tightly coupled with sensing, inference, control, and decision-making. Traditional communication paradigms centered on accuracy, throughput, and latency are increasingly inadequate for these systems, where the value of information depends on its semantic relevance to a specific task. This paper provides a unified exposition of the progression from classical distortion-based frameworks, through information freshness metrics such as the Age of Information (AoI) and its variants, to the emerging paradigm of goal-oriented semantics-aware communication. We organize and systematize existing semantics-aware metrics, including content- and version-aware measures, context-dependent distortion formulations, and history-dependent error persistence metrics that capture lasting impact and urgency. Within this framework, we highlight how these metrics address the limitations of purely accuracy- or freshness-centric designs, and how they collectively enable the selective generation and transmission of only task-relevant information. We further review analytical tools based on Markov decision process (MDP) and Lyapunov optimization methods that have been employed to characterize optimal or near-optimal timing and scheduling policies under semantic performance criteria and communication constraints. By synthesizing these developments into a coherent framework, the paper clarifies the design principles underlying goal-oriented, semantics-aware communication systems. It illustrates how they can significantly improve efficiency, reliability, and task performance. The presented perspective aims to serve as a bridge between information-theoretic, control-theoretic, and networking viewpoints, and to guide the design of semantic communication architectures for 6G and beyond.