<p>The theory of general fractional calculus with Sonine kernels has been well developed by Luchko in the one-dimensional case. Inspired by recent work on Mikusiński’s operational calculus for fractional partial differential operators, we construct a multi-dimensional version of the theory of Sonine kernels, solving a recognised open problem in the field. Starting from a generalised version of the classical Sonine convolution condition, we construct fractional integral and derivative operators in arbitrary dimensions, and examine their properties such as fundamental theorems of fractional calculus. Illustrative examples of the general theory are also included.</p>