Diffusion dominates mass transport in most electrochemical systems. In classical experimental systems on the micrometer scale or larger, this is adequately described at the mean-field level. However, nanoscale detection devices are being developed in which a handful or even single molecules can be detected. Brownian dynamics become manifest in these systems via the associated fluctuations in electrochemical signals. Here we describe the state of the art of these electrochemical nanodevices, paying particular attention to the role of Brownian dynamics and emphasizing areas in which theoretical understanding remains limited.