Abstract: This overview seeks to introduce, to a general mathematical audience, recent results and problems in several complex variables by surveying currently active topics such as the following.

 (i) Insolvable partial differential equation of Hans Lewy. Hörmander’s hypoellipticity of sums of squares of vector fields. Kohn’s subelliptic multipliers for weakly pseudoconvex domains with D’Angelo’s finite type condition.

 (ii) Roles of flat directions in global non-deformability of irreducible compact Hermitian symmetric manifolds and in strong and super rigidity problems.

 (iii) Effective theorems such as the Fujita conjecture, especially its very ampleness part.

 (iv) Skoda’s division and finite generation of canonical ring for any general compact complex manifold.

 (v) Hyperbolicity problem to lower degree required for hyperbolicity of generic complex hypersurface. Nevanlinna theory with counting function for nonsingular complex hypersurface in complex projective space.

 (vi) Gelfond-Schneider’s technique for Hilbert’s 7th problem. Lang-Bombieri’s theory of algebraic values of meromorphic maps. Bombieri-Pila’s counting of integral points on arcs. Pila-Wilkie’s o-minimal geometry, together with further later developments by many others, to treat the Andr´e-Oort conjec ture.