We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature; extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-��(M)$, where $��(M)$ is the Euler characteristic of $M$.

Complete revision. We've worked hard to make the text clearer and we hope the reader will notice the difference. Shorter and more conceptual proofs. The applications, we hope, are much clearer. In addition, we develop a new concept of "weakly smooth manifolds". This provides a nice smooth manifold structure for the space of unparametrised immersions (details in the introduction)