The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading nontrivial contributions to O (1/n) are derived for the two independent correlation exponents η_L2 and η _L4, and the related anisotropy index θ. The series coefficients of these 1/n corrections are given for general values of m and d with 0≤m≤d and 2+m/2<d<4+m/2 in the form of integrals. For special values of m and d such as (m,d) = (1,4), they can be computed analytically, but in general their evaluation requires numerical means. The 1/n corrections are shown to reduce in the appropriate limits to those of known large-n expansions for the case of d-dimensional isotropic Lifshitz points and critical points, respectively, and to be in conformity with available dimensionality expansions about the upper and lower critical dimensions. Numerical results for the 1/n coefficients of η_L2, η_L4 and θ are presented for the physically interesting case of a uniaxial Lifshitz point in three dimensions, as well as for some other choices of m and d. A universal coefficient associated with the energy-density pair correlation function is calculated to leading order in 1/n for general values of m and d.