There are two common ways of implementing ILU or (S)SOR preconditioned Krylov subspace methods on parallel computers. The rst resorts to multicoloring of the adjacency graph and often results in a parallelism of order N, where N is the dimension of the matrix. This strategy sometimes suuers from the deterioration of the rate of convergence. The second technique exploits the intrinsic parallelism available in the original forward and backward solutions. This approach does not suuer from the deterioration of the convergence rate since the resulting algorithm is mathematically equivalent to the original one. On the other hand, the maximum parallelism is limited by the length of the wavefronts which are often nonuniform. In this paper we consider the implementation of these two approaches on distributed memory computers and compare their performance on the CM-5. We have found that for the problems tested ILU(0) with multicoloring outperforms the other alternatives .