This paper investigates operators that commute with a discrete group of translations. Classes of such operators have received considerable attention in the context of approximation theory and shift-invariant systems, Gabor analysis, and wavelet frames. The authors extend the notion on shift-invariant operators on the Schwartz space of rapid descent functions \(S(\mathbb{R}^n)\) into the tempered distributions \(S'(\mathbb{R}^n)\). Since these operators are from the rapid descent functions into the tempered distributions, they are described by distributional kernels, and the shift-invariant operators can be treated as a special case of shift-invariant distributions. For \(x\in \mathbb{R}^n\), let \(T_x\varphi(t)= \varphi(t-x)\) denote the translation of a function on Euclidean \(n\)-space; the notion also extends to distributions. The authors prove a wonderful structure theorem for shift-invariant operators in the given setting, that is, for operators which commute only with a discrete subgroup of translations. The theorem proven is as follows: Theorem. Given \(a>0\), suppose that the operator \(A : S(\mathbb{R}^n)\to S'(\mathbb{R}^n)\) satisfies \(AT_{ak}=T_{ak}A\) for all \(k\in \mathbb{Z}^n\). Then, under this hypothesis, there exists a uniquely defined tempered distribution \(w_k\in S'(\mathbb{R}^n)\) for all integers \(k\), such that \(A\) can be written as a series of distributional coefficients on the translation operators. The operator \(A\) is continuous from the linear operators from the Schwartz space of rapid descent test functions to the Schwartz space of tempered distributions. The results are presented in a very neat and orderly manner.