We discuss the problem of positive-semidefinite extension: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I x I to a covariance kernel on the entire domain I x I. For a broad class of domains Ω called serrated domains, we present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We interpret the canonical completion via the graphical model structure it induces on the associated Gaussian process. Furthermore, we show how the determination of the canonical completion reduces to the solution of a system of linear inverse problems in the space of Hilbert-Schmidt operators, and derive rates of convergence when the kernel is to be empirically estimated. We conclude by providing extensions of our theory to more general forms of domains, and by demonstrating how our results can be used in statistical inverse problems associated with stochastic processes.[-]