Semiconductor nano-lasers have been actively investigated both theoretically and experimentally with to the aim of providing a highly compact laser amenable to photonic integration. Such devices are naturally suited for assembly in close-packed one- and two-dimensional arrays. In such arrangements, optical coupling between elements of the array opens opportunities to generate a range of dynamical behaviours. In this paper, we present the first theoretical treatment of the dynamics of electrically pumped nano-laser arrays. Two specific forms of such arrays are analysed in detail: a three-element linear array, and triangular arrays. The former is the basis for extensive one-dimensional arrays, whilst the latter is a building block of many possible geometric configurations of two-dimensional nanolaser arrays. Using these prototypical configurations enables the identification of novel dynamical behaviours, which may be accessed using nano-laser arrays. A distinguishing physical feature of nano-lasers is the enhancement of the spontaneous emission rate via the so-called Purcell effect. Allowing for a range of Purcell enhancement factors, the analysis focusses on the effects of experimentally controllable parameters such as the laser drive current. It is shown that the Purcell enhancement factor is critical to the availability of a range of dynamical behaviours which arise simply due to inter-element optical coupling. Two-dimensional portraits of the regimes of differing dynamics offer a convenient means for determining the dynamics which may be accessed by varying the laser drive current.