The design of analogue electronic experiments to investigate phenomena in nonlinear dynamics, especially stochastic phenomena, is described in practical terms. The advantages and disadvantages of this approach, in comparison to more conventional digital methods, are discussed. It is pointed out that analogue simulation provides a simple, inexpensive, technique that is easily applied in any laboratory to facilitate the design and implementation of complicated and expensive experimental projects; and that there are some important problems for which analogue methods have so far provided the only experimental approach. Applications to several topical problems are reviewed. Large rare fluctuations are studied through measurements of the prehistory probability distribution, thereby testing for the first time some fundamental tenets of fluctuation theory. It has thus been shown for example that, whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced by added noise, has been widely studied by analogue methods, and the main results are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise-induced linearization are also described. Selected examples of the use of analogue methods for the study of transient phenomena in time-evolving systems are reviewed. Analogue experiments with quasimonochromatic noise, whose power spectral density is peaked at some characteristic frequency, have led to the discovery of a range of interesting and often counter-intuitive effects. These are reviewed and related to large fluctuation phenomena. Analogue studies of two examples of deterministic nonlinear effects, modulation-induced negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR) are described. Finally, some speculative remarks about possible future directions and applications of analogue experiments are discussed.