We give a complete description of the linear dynamics of multiplication Mm and composition operators Cφ on the space Hol(D) of all holomorphic maps on the unit disc. We show that Mm is never supercyclic, and cyclic if and only if the map m is injective. For composition operators, we prove that if φ has a fixed point in D, then Cφ is either not cyclic, or cyclic but not supercyclic on Hol(D). On the other hand, if φ does not have any fixed point in the unit disc, then Cφ is hypercyclic on Hol(D). We provide explicit expressions of cyclic and hypercyclic vectors. Finally, we make some observations on weighted composition operators on Hol(D).