How do we describe the topology of the space of all nonconstant holomorphic (respectively, algebraic) maps F: X--- greater than Y  from one complex manifold (respectively, variety) to another? What is, for example, its cohomology? Such problems are old but difficult, and are nontrivial even when the domain and range are Riemann spheres. In this talk I will explain how these problems relate to other parts of mathematics such as spaces of polynomials, arithmetic (e.g the geometric Batyerv-Manin type conjectures), algebraic geometry (e.g. moduli spaces of elliptic fibrations, of smooth sections of a line bundle, etc) and if time permits, homotopy theory (e.g. derived indecomposables of modules over monoids). I will show how one can fruitfully attack such problems by incorporating techniques from topology to the holomorphic/algebraic world (e.g. by constructing a new spectral sequence).