Some problems of selection and ranking for the multivariate normal populations are studied. The major part of the paper (Section 2) deals with the selection problem in terms of the population multiple correlation coefficient. Both unconditional and conditional cases are studied for the largest (smallest) multiple correlation. Selection procedures R1, R2, R3, and R4 are proposed for the largest multiple correlation case while procedures R5, R6, R7, and R8 are proposed for the case of the smallest. Asymptotic results are obtained. Properties of the selection procedures are investigated. Sufficient conditions are obtained for the monotonicity of certain probability integrals in terms of the non-centrality parameter. Which is involved in the negative binomial weights (Theorem 2.6). Tables of the percentage points of the statistics which give appropriate constants for procedures R1, R2, R3, R4, R7, and R8 are constructed and appended at the end. Section 3 deals with the selection of p-variate normal populations. When the variables are partitioned into two sets of q1 and q2 components, the criterion of ranking being the generalized conditional variance of the q2-set (q1-set fixed). (Author).