Abstract: In classical algebraic and differential topology cobordism and associated invariants play a central role. In symplectic topology, Lagrangian cobordism has been explored starting with ideas of Arnold followed by work of Eliashberg and Audin and later of Chekanov. The general impression was that this is a very ``soft'' notion - and thus of limited interest - as it reduces to immersed cobordism and, in turn, the immersed notion is ``homotopical'' by an application of the $h$-principle. In this talk based on recent work with Paul Biran I will discuss some new results on this topic: it turns out that that most Floer type invariants are in fact left invariant by appropriate but quite general cobordisms - so Lagrangian cobordism is much more rigid with respect to these algebraic structures than expected. This has wide-reaching implications and I hope to discuss some of them in the talk.