This article grew out of several talks that the author presented at the Banach Institute and at the University of Bia lystok in Poland during November of 2001. It describes six problems from the geometry of submanifolds. Some of the problems come from the theory of constant curvature submanifolds in Euclidean space, as well as applications of Morse theory of the height function to the problem of relating curvature and topology of submanifolds in Euclidean space. Others come from infinite-dimensional Morse theory of minimal surfaces in Riemannian manifolds.