We derive a Weierstrass-type formula for conformal Lagrangian immersions in complex Euclidean 2-space, and show that the data satisfies a Dirac-type equation with complex potential. We apply the representation to the Hamiltonian-stationary case to construct all possible tori and obtain a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. Results described here are contained in two articles in collaboration with Frédéric Hélein.