Abstract We prove that Hilbert’s Tenth Problem for meromorphic functions and for entire functions in several variables is unsolvable over the language of rings, together with constant symbols for two of the variables and a predicate for a place. This is the first result in the literature, which proves unsolvability of a diophantine problem for the ring of complex analytic functions in a number of variables. The proof rests upon an interplay between analytic geometry, analysis, number theory, and logic.