Is there any work on manifolds (perhaps contact) with symplectic boundary (not asking about the boundary of a symplectic manifold)?
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There is a book by Guillemin, Gizburg and Karshon: Moment Maps, Cobordisms, and Hamiltonian Group Actions. 


For contact manifolds there is the notion of a convex hypersurface due to Giroux, ie, a hypersurface that is transverse to a contact vector field. A convex hypersurface is composed by two exact symplectic domains (not necessarily connected) glued together along the dividing set (the dividing set is a hypersurface in the hypersurface) that carries a natural contact structure (see also this question). I don't know if this helps, since the boundary is not a closed symplectic manifold, but it is a very natural object to consider in contact topology. 

