In the first part of this talk I will describe a symplectic approach to K¨ahler geometry on toric orbifolds. The main result is an effective parametrization of all toric K¨ahler metrics via smooth functions on the moment polytope. In the second part of the talk I will give an application to the description of U(n)- invariant extremal K¨ahler metrics. Through this symplectic approach these arise naturally as solutions to a linear second order ODE.