A Taylor complex is a graded collection of polynomials satisfying a few simple axioms. We show that there is a natural one-to-one correspondence between the solution sets of linear constant coefficient differential equations and Taylor complexes. Using this correspondence, we then give a characterization of the solution sets of linear constant coefficient differential equations, which follows closely Willems' characterization of the solution sets of linear constant coefficient difference equations. We also characterize linear differential operators. (The presentation is given for the case of several variables.)