Stacked intelligent metasurfaces (SIMs) cascade several closely spaced programmable metasurface layers so that an electromagnetic wave undergoes repeated modulation-propagation interactions before leaving the stack. The stack thereby implements, at the speed of propagation and with few radio-frequency chains, high-dimensional analog transformations that would otherwise require digital precoding, combining, spatial filtering, focusing, or inference. Since 2023, SIM research has produced more than two hundred articles, a dedicated magazine special issue, and two survey-level overviews organized around applications and wave-domain computing. This article complements them with a model-to-hardware perspective. First, we present the electromagnetic modeling hierarchy of SIMs, making explicit the assumptions under which the models coincide, their validity regimes, and their computational complexity, with a common notation for hardware non-idealities and power consumption. Second, we present a methodology-first taxonomy (SIM-aware channel acquisition, model-based optimization, statistical-CSI design, learning-based control, and hardware-aware robust design), compare the families in terms of CSI requirements, complexity, convergence, and hardware constraints, and reconcile conflicting findings in the literature. Third, we synthesize applications by the physical role of the SIM. Fourth, we quantitatively compare experimentally demonstrated multilayer prototypes, separate validated from conceptual architectures, and discuss thermal, tolerance, and calibration issues. Fifth, we provide a tutorial-style performance and trade-off analysis anchored in reported results. Finally, we map evidence-based challenges to research directions with measurable targets. Lessons learned are summarized at the end of every section.

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