Let $(X,H)$ be a polarized algebraic surface. Let $M=M^H_X(c_1,c_2)$ be the moduli spaces of $H$-semistable rank 2 sheaves on $X$ with Chern classes $c_1, c_2$.  K-theoretic Donaldson invariants of $X$ are holomorphic Euler characteristics of determinant line bundles on $M$. These invariants are subject to wallcrossing when $H$ varies. In the first part of the lecture I present joint work with Nakajima and Yoshioka, where we determine a generating function for the wallcrossing in terms of elliptic functions. If time permits I will in the second part of the talk present some results about the K-theoretic invariants of rational surfaces, and relate these to Le Potiers strange duality conjecture.