Mobile devices increasingly require the parallel execution of several computing tasks offloaded at the wireless edge. Existing communication systems only support parallel transmissions at the bit level, which fundamentally limits the number of tasks that can be concurrently processed. To address this bottleneck, this paper introduces the new concept of Semantic Multiplexing. Our approach shifts stream multiplexing from bits to tasks by merging multiple task-related compressed representations into a single semantic representation. As such, Semantic Multiplexing can multiplex more tasks than the number of physical channels without adding antennas or widening bandwidth by extending the effective degrees of freedom at the semantic layer, without contradicting Shannon capacity rules. We have prototyped Semantic Multiplexing on an experimental testbed with Jetson Orin Nano and millimeter-wave software-defined radios and tested its performance on image classification and sentiment analysis while comparing to several existing baselines in semantic communications. Our experiments demonstrate that Semantic Multiplexing allows jointly processing multiple tasks at the semantic level while maintaining sufficient task accuracy. For example, image classification accuracy drops by less than 4% when increasing from 2 to 8 the number of tasks multiplexed over a 4$\times$4 channel. Semantic Multiplexing reduces latency, energy consumption, and communication load respectively by up to 8$\times$, 25$\times$, and 54$\times$ compared to the baselines while keeping comparable performance. We pledge to publicly share the complete software codebase and the collected datasets for reproducibility.