Jessup earned a BFA in graphic design at OSU in 1976. After receiving an MFA from Stanford in 1978, he worked for Korty Films, Lucasfilm, and Industrial Light and Magic. While at ILM, he received an Academy Award for special effects work on the film Innerspace and a nomination for Hook. In 1996 he joined Pixar, where he worked on many of successful films including Toy Story 2, Monsters, Inc. and Ratatouille. For the latter he received an ANNIE award for Production Design in an Animated Feature.