This paper is concerned with the algebraic and analytic structure of the commutant g’(L) of a regular ordinary differential operator L with Cz matrix-valued coefficients. The main algebraic result is that V(L) is a free module of rank at most EW over the polynomial ring U&L], where K is the six of the matrices in the coefficients of L and n is the order of L. This result, along with seseral related results, is first proved in a completely algebraic setting, namely for differential operators whose coefficients are matrices over a commutative differential ring in which systems of homogeneous linear differential equations have finite-dimensional solution sets. In the Cm case, the algebraic structure of V(L) is obtained from an embedding of %?(A) into the ring of ek :: nk matrices over a polynomial ring @[A], and the image of Q(L) in this matrix ring is completely determined. If %YJL) denotes the set of those operators in q(L) with rank at most nz, then it is shown that the dimension of V,,(L) is an upper semicontinuous function of L. The paper concludes by finding some first integrals for the commutation equation L T TL = 0. Our study of cornmutants of regular ordinary differential operators is predated by the work of Amitsur [l], Burchnall and Chaundy :3], and Krichever [6]. Amitsur‘s work is completely algebraic, but is restricted to the case of a derivation acting on a field of characteristic zero. Burchnall and Chaundy restrict themselves to operators with scalar-valued coefficients. Krichevcr allows matrixvalued coefficients, but restricts himself to the study of commuting pairs L and T both of whose leading coefficients must be constant nonsingular diagonal matrices. In [6], commuting pairs L and T are analyzed by letting T act on certain eigenfunctions of L. These eigenfunctions have a formal series representation, and much information is gleaned from the coefficients of the series. The action of T on eigenspaces of L is also basic to our work, but our approach is more closely related to an idea appearing in IS]. We choose a convenient basis for the eigenspaces of L and study the matrices representing T on these eigenspaces with respect to the “nice” basis. In both [3] and [6] it was noticed that if T and L commute, then there is a nonzero polynomial P(A., p) in two commuting variables such that P(L, T) = 0.