Why Lagrangian cobordism? There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "Lagrangian" condition. For example, Cheptea-Habiro-Massuyeau consider a category whose morphisms are cobordisms $M$ between closed oriented surfaces $F_+$ and $F_-$, where we choose Lagrangian subgroups $A_{\pm}$ of $H_1(F_\pm)$  correspondingly, and where we require that $H_1(M)=m_-(A_-)+m_+H_1(F_+)$ and that $m_+(A_+)\subseteq m_-(A_-)$ in $H_1(M)$ (the $m_\pm$ are inclusion maps). Similar conditions are imposed in many other papers.
I have never understood why such conditions are imposed. One half-thought I have is that it is related to Wall's result that the kernel of the inclusion of $H_1(\partial M)$ in $H_1(M)$ is a Lagrangian subgroup of $H_1(\partial M)$. Another half-idea is that it might be some weak 2-framing condition for the cobordism or something.

What is the conceptual explanation for this Lagrangian condition and its variants? Does it have anything to do with framing? (or orientation?) Why do "symplectic" and "Lagrangian" have anything to do with TQFT? (is it all just Wall's result in some guise?)

Every time I see a Lagrangian condition I feel very stupid for not knowing what it's doing there.
 A: A historical  reason for the importance of things Lagrangian in a TQFTy setting    comes from Floer Homology and a conjecture of Atiyah that ``the two'' Floer theories are the same for 3-manifolds. Floer defined one homology (monopole Floer homology) based on the Chern Simons action functional, with the generators of its chain complex (the analogues of critical points in Morse Theory) being flat connections.  He built another  homology theory (the more standard one, today) to deal with the Arnol'd conjecture in Hamiltonian mechanics.  If you take a Heegegard splitting of the 3-manifold, then, associated to  the dividing surface you have  the moduli space of flat connections, which is a symplectic manifold. You can then start to set up a Hamiltian Floer theory on this moduli space.    Those flat connections over the surface which extend into the three-manifold on one side of the surface define a Lagrangian submanifold of the moduli space.  Those flat connections which extend to the other side define another Lagrangian submanifold.  The  symplectic Floer business on the moduli space, set up properly,   gives you a count the intersections of these two Lagrangian submanifolds, and hence of the set  flat  connections which extend to the whole 3-manifold.  Atiyah conjectured these two Floer theories  are the same.
I suggest as an entry the   Math Reviews article, See MR1283871,  by Donaldson on the proof of a special case of this conjecture by  Deitmar Salamon and Stamatis Dostoglou.
A philosophical reason is Weinstein's "symplectic creed": everything is Lagrangian.
The classical version of a quantum state is a Lagrangian submanifold.  So if one's
TQFT yields a "semi-classical" Hilbert space, say a symplectic vector space, then the states (wavefunctions) of the "Hilbert space" be   Lagrangian subspaces 
