The development and testing of alternative numerical methods and computational algorithms specifically designed for the vectorization of transport and diffusion computations on a Control Data Corporation (CDC) Cyber 205 vector computer are described. Two solution methods for the discrete ordinates approximation to the transport equation are summarized and compared. Factors of 4 to 7 reduction in run times for certain large transport problems were achieved on a Cyber 205 as compared with run times on a CDC-7600. The solution of tridiagonal systems of linear equations, central to several efficient numerical methods for multidimensional diffusion computations and essential for fluid flow and other physics and engineering problems, is also dealt with. Among the methods tested, a combined odd-even cyclic reduction and modified Cholesky factorization algorithm for solving linear symmetric positive definite tridiagonal systems is found to be the most effective for these systems on a Cyber 205. For large tridiagonal systems, computation with this algorithm is an order of magnitude faster on a Cyber 205 than computation with the best algorithm for tridiagonal systems on a CDC-7600. The above-mentioned algorithm for solving tridiagonal systems is also used as a basis for a new hyperline method for implementing the red-black cyclic Chebyshev iterative method to the solution of two-dimensional diffusion problems. The hyperline method is found to be competitive with other alternative options. This hyperline method has an attractive feature of being compatible with so-called concurrent iteration procedures whereby iterations n + 1,…, n + k, can be started before the completion of iteration n. This feature is very effective in balancing computations and data transfer requirements for very large diffusion problems. Consequently, the hyperline method is suitable for implementation for the solution of three-dimensional diffusion problems.