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. 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.