The aim of this article is the computation of topological invariants of Hitchin moduli spaces \(H(X,L,r,e)\), in particular the Poincaré and Hodge polynomials. The spaces \(H(X,L,r,e)\) are moduli spaces of sheaves with rank \(r\) and degree \(e\) over a genus \(g\) curve \(X\), together with a morphism \(\Phi:E\to E \otimes_X L\) where \(L\) is a coefficient line bundle. The authors present a recursive procedure, which determines the desired invariants for arbitrary \((r,e)\) and \(g\). The procedure is based on the wall-crossing of ADHM sheaves, which are sheaves supported on the non-compact Calabi-Yau manifold \(Y\) which is isomorphic to the total space of two line bundles \(M_1^{-1} \oplus M_2^{-1}\) over \(X\) with \(M_1\otimes_X M_2\cong K_X^{-1}\). The authors show that the invariants of Hitchin moduli spaces appear in wall-crossing formulas for the invariants of moduli spaces of these ADHM sheaves (Donaldson-Thomas invariants) upon varying the stability condition. The Donaldson-Thomas invariants can be computed in an asymptotic chamber in terms of sums over Young tableaux using geometric engineering and the topological vertex formalism. As a result, the wall-crossing formula provides a recursive formula for the computation of invariants of Hitchin moduli spaces. The authors list explicit Poincaré and Hodge polynomials for small rank, degree and genus, and find agreement with earlier results in the literature by Hitchin, Gothen, Hausel and Rodriguez-Villegas. The recursive formula was solved more recently by S. Mozgovoy [“Solutions of the motivic ADHM recursion formula”, aXiv:1104.56982011].