Pulsed-laser deposition has been used to grow epitaxial thin films of the giant-dielectric-constant material CaCu3Ti4O12 on LaAlO3 and SrTiO3 substrates with or without various conducting buffer layers. The latter include YBa2Cu3O7, La1.85Sr0.15CuO4+δ, and LaNiO3. Above 100–150 K, the thin films have a temperature independent dielectric constant as do crystals. The value of the dielectric constant is of the order of 1500 over a wide temperature region, potentially making it a good candidate for many applications. The frequency dependence of its dielectric properties below 100–150 K indicates an activated relaxation process. © 2002 American Institute of Physics.