Lagrangian translating solitons are Lagrangian submanifolds that evolve by translations under mean curvature flow, and they are natural candidates for the local models of finite-time singularities. In this talk, I will describe a gluing construction of Lagrangian translating solitons, which allows us to construct new examples with complicated geometries. The basic idea is to first build approximate solutions by gluing in Lawlor necks to the intersection points of Grim Reapers and then solve the nonlinear PDE near the approximate solutions by a fixed point argument. The main difficulty is the invertibility of the linearized operator, which we get round by introducing suitable weighted Sobolev spaces on the approximate solutions.