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We determine a nice simple formula for the largest Euclidean space for whichthere is an orientable n-manifold with a nonimmersion detected byStiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upperbound and establish the complete answer for n<24 and n=33,34. Results similarto many of these were obtained some 50 years ago, but in a much less tractableform. The sharp results for Spin manifolds require detailed calculations ofko-homology groups of mod-2 Eilenberg-MacLane spaces.

Stiefel-Whitney classes and immersions of orientable and Spin manifolds