I will present a recent result (joint with Yakov Eliashberg) demonstrating the existence of exact Lagrangian cobordisms with a loose Legendrian in the negative end, in all dimensions greater than 4. In particular we show that there exists a Lagrangian disk in Cn \ B2n which has Legendrian boundary in S2n-1, whenever n>2. It is known there are no such disks in C2. The proof showcases a new “Lagrangian Whitney trick”. As an application, we prove a universal embedding theorem for flexible Weinstein manifolds, showing in particular that any Weinstein manifold has interesting geometry only in a topological collar of the boundary. We also construct exact Lagrangian immersions with fewer intersections than the “philosophy of the Arnold conjecture” would predict (this application is joint with Tobias Ekholm, Eliashberg, and Ivan Smith).