This paper presents a study of the best and worst mappings for the omega network proposed by D. H. Lawrie in 1975. We identify mappings that produce no conflicts in the network and mappings that produce a maximum number of conflicts. The analysis of mappings for some typical applications shows that an initial allocation of data to memory modules determines the contention within the network for all iterations of the algorithm. For the case of the FFT and the bitonic sort algorithm executed on a shared-memory architecture, we prove that if no conflicts are produced during the first iteration of the algorithm, then no conflicts are produced during any other iteration. Moreover, if a maximum number of conflicts are produced during the first iteration, then a maximum number of conflicts are produced during all other iterations of the algorithm. For the d-dimensional grid computations where communication is required with 2d nearest neighbors, we prove that if the initial allocation produces no conflicts within the network, then communication with all the neighbors is conflict-free. Also, if the initial allocation produces a maximum number of conflicts, then communication with all the neighbors is maximum-conflict. We show that the omega network cannot produce without conflicts some of the bit-permute mappings such as the perfect shuffle and the bit reversal. The network can produce both of these mappings provided that data items are accessed from memories according to a specific skewed scheme.