Holomorphic mappings are basic to Complex Analysis. Approximation theory is at the heart of holomorphic mapping theory. The state of the art in this area is the so called Andersen-Lempert the- ory for approximation of injective holomorphic maps. The theory, which emerged in the 1990s, has been an immense success, and has found a huge number of applications in a variety of topics, including, but not limited to, the topics mentioned above. However, the theory requires strong geometric constraints, and is limited to the equi-dimensional setting, and this leaves us with major challenges for further advances. We expect that bringing together experts from around the world, in or- der to discuss promising new directions will lead to many new collaborations and results on fundmental questions such as on the injective Kobayashi metric, complex dynamics of entire holomorphic automorphisms, uniformization theory and Loewner theory.