"The goal of this project is to study the topology of Stein manifolds from the viewpoint of symplectic and contact geometry. It addresses the fundamental questions of the subject: - How does the Lagrangian skeleton of a Stein manifold determine the Stein structure? - To what extent the study of Stein structures can be reduced to a combinatorial study of the skeleton? - How are the symplectic invariants of Stein manifolds, respectively the contact invariants of their boundary, determined by the skeleton? For the topological part, we will use as a source of inspiration the theory of spines and shadows of 3- and 4- manifolds. One of the goals of this research project is to adapt it to the setup of Stein manifolds and develop a calculus of Lagrangian shadows. Concerning invariants of contact manifolds, we aim to interpret symplectic homology of Stein manifolds and contact homology of their boundaries in topological terms, with the skeleton playing a central role. Further ramifications of this research project include the development of string topology on singular (stratified) spaces and the symplectic study of singularities."