In this seminar, I shall begin reviewing the wall-crossing phenomena in four-dimensional N=2 supersymmetric gauge theories, and the recent developments in encoding the BPS spectra from the moduli spaces of closely related compactified theories on R³ x S¹. The Konsevich-Solibelman wall-crossing formula can then be deduced from the physical smoothness condition of the moduli space metric. In remaining time, I will describe the explicit semi-classical instanton computations in the compactified gauge theories, which constitute a non-trivial test for the conjectured quantum corrected moduli space, and the underlying wall-crossing formula. If time permits, I will also describe some ongoing work on explicit multi-instanton computation for the higher rank (>1) gauge theories.