  In this article we define Lagrangian concordance of Legendrian knots, the
analogue of smooth concordance of knots in the Legendrian category. In
particular we study the relation of Lagrangian concordance under Legendrian
isotopy. The focus is primarily on the algebraic aspects of the problem. We
study the behavior of the classical invariants under this relation, namely the
Thurston-Bennequin number and the rotation number, and we provide some examples
of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using
these examples, we are able to provide a new proof of the local Thom
conjecture.
