Measurement of the electric dipole moment of a fundamental particle is one way in which to search for physics beyond the Standard Model. In this talk I will review current experiments which set limits on the electric dipole moment (EDM) of the electron. All these experiments use heavy polar molecules. I will concentrate in particular on the experiment at Imperial College which uses YbF (ytterbium fluoride). The experiment is based on electron spin precession in a strong electric field. I will discuss the measurement technique and how systematic effects are controlled. I will also explain how new techniques of directly laser cooling of molecules have increased the sensitivity of this experiment, and how using these techniques will lead to an ever larger sensitivity increase in the near future.