  We call a closed, connected, orientable manifold in one of the categories
TOP, PL or DIFF chiral if it does not admit an orientation-reversing
automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly
chiral if it does not admit a self-map of degree -1. We prove that there are
strongly chiral, smooth manifolds in every oriented bordism class in every
dimension greater than two. We also produce simply-connected, strongly chiral
manifolds in every dimension greater than six. For every positive integer k, we
exhibit lens spaces with an orientation-reversing self-diffeomorphism of order
2^k but no self-map of degree -1 of smaller order.
