Abstract: The classical physics and mechanics in flat space are first reviewed. Three typical examples of physics and mechanics in curved space, i.e. the biomembrane mechanics, the shallow shell mechanics and the diffusion dynamics on curved surfaces, are explained. The essential differences between the physics and mechanics in flat space and curved space are discussed. It is shown that there is only one fundamental differential operator in the physics and mechanics in flat space, i.e. the classical gradient. However, there are two fundamental differential operators in the physics and mechanics in curved space, i.e. the classical gradient and the shape gradient.