The paper contains two comparison theorems for vector-valued stochastic differential equations. The first result deals with the finite- dimensional case. Let us agree that vectors \(x,y\) in \(R^ d\) are ordered, that is, \(x<y\) if this inequality holds coordinate-wise. Let \(X^ i\) be a solution to the equation \(dX^ i = F_ i(X^ i) dt + G(X^ i) dW\) in \(R^ d\) for \(i=1,2\). Then under some technical assumptions it is shown that if \(F_ 1<F_ 2\) and \(X_ 1(0) < X_ 2(0)\), then for all \(t>0\) we have \(X_ 1(t)<X_ 2(t)\). Next, a pair of stochastic partial differential equations \[ du_ i = \left( {\partial^ 2 u_ i \over \partial x_ i^ 2} + f_ i(u_ i) \right) dt + g(u_ i)dW \] in a bounded interval is considered, where \(W\) is a cylindrical Brownian motion. The solution to this equation is defined in the sense of Walsh. It is shown that the comparison theorem holds for this pair of equations. The proof rests on the discretisation of space variable which leads to a finite-dimensional stochastic differential equation. Then the first result of the paper can be applied to obtain the comparison theorem for the approximating equation and finally a certain limiting argument ends the proof.