The sixth generation (6G) network is expected to deploy larger multiple-input multiple-output (MIMO) arrays to support massive connectivity, which will increase overhead and latency at the physical layer. Meanwhile, emerging 6G demands such as immersive communications and environmental sensing pose challenges to traditional signal processing. To address these issues, we propose the ``semantic-aware MIMO'' paradigm, which leverages specialist models and large models to perceive, utilize, and fuse the inherent semantics of channels and sources for improved performance. Moreover, for representative MIMO physical-layer tasks, e.g., random access activity detection, channel feedback, and precoding, we design specialist models that exploit channel and source semantics for better performance. Additionally, in view of the more diversified functions of 6G MIMO, we further explore large models as a scalable solution for multi-task semantic-aware MIMO and review recent advances along with their advantages and limitations. Finally, we discuss the challenges, insights, and prospects of the evolution of specialist models and large models empowered semantic-aware MIMO paradigms.