Advances in machine learning technology have enabled real-time extraction of semantic information in signals which can revolutionize signal processing techniques and improve their performance significantly for the next generation of applications. With the objective of a concrete representation and efficient processing of the semantic information, we propose and demonstrate a formal graph-based semantic language and a goal filtering method that enables goal-oriented signal processing. The proposed semantic signal processing framework can easily be tailored for specific applications and goals in a diverse range of signal processing applications. To illustrate its wide range of applicability, we investigate several use cases and provide details on how the proposed goal-oriented semantic signal processing framework can be customized. We also investigate and propose techniques for communications where sensor data is semantically processed and semantic information is exchanged across a sensor network.