Next-generation wireless networks are expected to utilize the limited radio frequency (RF) resources more efficiently with the aid of intelligent transceivers. To this end, we propose a promising transceiver architecture relying on stacked intelligent metasurfaces (SIM). An SIM is constructed by stacking an array of programmable metasurface layers, where each layer consists of a massive number of low-cost passive meta-atoms that individually manipulate the electromagnetic (EM) waves. By appropriately configuring the passive meta-atoms, an SIM is capable of accomplishing advanced computation and signal processing tasks, such as multiple-input multiple-output (MIMO) precoding/combining, multi-user interference mitigation, and radar sensing, as the EM wave propagates through the multiple layers of the metasurface, which effectively reduces both the RF-related energy consumption and processing delay. Inspired by this, we provide an overview of the SIM-aided MIMO transceiver design, which encompasses its hardware architecture and its potential benefits over state-of-the-art solutions. Furthermore, we discuss promising application scenarios and identify the open research challenges associated with the design of advanced SIM architectures for next-generation wireless networks. Finally, numerical results are provided for quantifying the benefits of wave-based signal processing in wireless systems.