Smooth cobordisms between submanifolds play an important role in topology. I will discuss smooth cobordisms that have additional geometric conditions imposed by symplectic and contact structures. In particular, I will discuss Lagrangian cobordisms between Legendrian knots. Whereas the genus of a smooth cobordism (orientable or not) between a knot and itself can be arbitrary, the genus of an orientable Lagrangian endocobordism must vanish, and non-orientable Lagrangian endocobordisms must have a genus that is a multiple of 4. Furthermore, some Legendrian knots will have an infinite number of “exact” non-orientable, Lagrangian endocobordisms, while other Legendrian knots have none. This is joint work with Orsola Capovilla-Searle.