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Analog computing is based upon using physical processes to solve formal mathematical problems. In the past, it was the predominant instrument of scientific calculations. Now, as the physical limits imposed on digital devices compel research into alternate computing paradigms, a reexamination of the potentialities of analog computing is warranted. This work studies the application of analog CMOS cells toward the simulation of dynamical systems, and, more generally, solving sets of coupled time-dependent ordinary differential equations. Following a brief review of the fundamentals of systems theory and analog computing, the main set of computing elements is introduced, each comprising analog cells designed in a 130 nm process. These are subsequently applied to the realization of practical, special-purpose analog computing modules. Illustrative systems from various fields are selected for simulation. Though by no means comprehensive, these case studies highlight the capabilities of contemporary analog computing, especially in solving nonlinear problems. Circuit simulations show good agreement with solutions obtained from high-order numerical methods, at least over a limited range of system parameters. The article concludes with a brief discussion of broader analog computing applications, offering future prospects toward further exploration of its potentialities and limitations in a wide range of domains.