The activity of neural populations can be well-described by relatively few population-wide activity patterns spanning a “neural manifold”. Virtually all these studies have analysed flat neural manifolds to understand how the brain controls behaviour. We hypothesised that since neurons have nonlinear responses and make thousands of recurrent connections that may enhance this nonlinearity, nonlinear manifolds should capture the neural population activity better than flat manifolds. Analysis of a centre-out reaching task in monkeys confirmed that, even during a relatively simple behaviour, motor cortical (MC) population activity is best captured by a nonlinear manifold. To investigate if manifold nonlinearity arises due to the dense connectivity patterns of brain circuits, we trained neural networks with different degrees of recurrent connectivity to perform this task. Indeed, manifold nonlinearity increased monotonically with the number of recurrent connections. To test in vivo this presumed influence of circuit connectivity on manifold nonlinearity, we compared neural manifolds from two anatomically distinct motor regions –MC and striatum– using simultaneous recordings from mice performing a grasping and pulling task. Manifold nonlinearity was strongly region-dependent: Striatal manifolds were consistently more nonlinear than MC manifolds. Besides circuit connectivity, we also expected task complexity to influence manifold nonlinearity. We hypothesised that if manifolds are nonlinear, more varied tasks requiring a richer set of neural activity patterns should reveal greater nonlinearities. We confirmed this prediction using neural population recordings from human MC during attempted handwriting. Drawing lines of varying length across 16 directions and writing all the letters in the English alphabet had more nonlinear manifolds than the simpler tasks of drawing lines in one direction or writing a handful of morphologically similar letters, respectively. Thus, manifolds underlying neural population activity during behaviour are nonlinear, their degree of nonlinearity depends on the connectivity of the brain region, and increases during more complex tasks.