Stacked intelligent metasurfaces (SIMs) facilitate computation by cascaded programmable layers so that part of the signal processing can be performed in the wave domain during signal propagation, rather than solely after reception. This approach expands the controllable degrees of freedom and supports the joint design of communication, sensing, and computation with the potential for reduced energy usage, shorter end-to-end latency, and improved task execution. Despite these advances, research on the SIM concept is still at an early stage, with challenges in scalability, controllability, nonlinearity, and robustness. This article reviews the state-of-the-art of SIM research, including applications, functions, and characteristics. We also demonstrate their potential through case studies on neural-like analog inference and communication enhancement. Finally, the paper outlines open challenges and future directions toward establishing SIMs as a new signal processing paradigm for in-wave computation in next-generation (NG) networks.