We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, the situation is more delicate. We will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings for which the set points where the map has degenerate rank is of codimension at least 2, is transversal to the target. In addition, we show that under more restrictive conditions on the manifolds, finite holomorphic mappings are transversal.