Hilbert's Tenth Problem in its original form is the following: Is there an algorithm that determines, given a multivariate polynomial equation with integer coefficients, whether the equation has a solution over the integers?

Davis-Putnam-Robinson-Matiyasevich showed that there is no such algorithm, i.e., Hilbert's Tenth Problem is undecidable. Since then the analogue of this question has been studied for various rings. I will discuss the result by Kim and Roush that Hilbert's Tenth Problem for C(t,s) is undecidable, and show how their result can be generalized to finite extensions of C(t,s), i.e., to function fields of complex surfaces.