Let A be a square matrix with rational entries. Then A can be written as the sum S + N where S and N also have rational entries, S is semisimple, N is nilpotent, and S and N commute. This representation is unique, and is called the Jordan decomposition of A. It can be considered as a coordinate-free, and ambient-field-free, version of the usual Jordan canonical form for matrices. This decomposition (in its multiplicative version) is particularly useful in the study of algebraic groups. If G is a finite group and a is an element of the rational group ring Q[G], i.e. a is a linear combination of group elements with rational coefficients, then there is an analogous decomposition: a = s + n where s and n lie in Q[G], s is semisimple, n is nilpotent, and s and n commute (this representation is also unique). Consider the integral versions of these decompositions: if the matrix A has integer entries, need S and N have integer entries? if the element a in Q[G] has integer coefficients, need s and n have integer coefficients? We give complete answers to these questions. The multiplicative version of the integral group ring question is much more subtle, however, and we only have partial results on this problem.