A number of studies have elucidated cases of the insufficiency of the linear noise approximation in biological contexts by comparison of its predictions with those of stochastic simulations. This has led to the investigation of higher order terms of the system size expansion that go beyond the linear approximation. These terms have been used to obtain more accurate moment estimates for the mean concentrations and for the variances of the concentration fluctuations in intracellular pathways. In particular, the leading order corrections to the linear noise approximation yield corrections of the conventional rate equations. Terms of higher order have also been used to obtain corrections to the variances and covariances estimates of the linear noise approximation. The linear noise approximation and corrections to it can be computed using the open source software intrinsic Noise Analyzer. The corrections have been shown to be particularly considerable for allosteric and non-allosteric enzyme-mediated reactions in intracellular compartments.