Two possible ways of extending Symanzik's improvement programme to lattice fermions namely improvement to first and second order in the lattice sppacing a are discussed. The corresponding lattice actions for fermions are constructed and tree-level improvement conditions are derived by considering "classical" improvement. The concept of "on-shell" improvement is generalized to the lattice fermions studied here and the free parameters are determined for O( a) and O( a2) on-shell improved actions to all orders of perturbation theory. No evidence is found that the complicated structure of the O( a2) on-shell improved action, especially thearising fermion contact terms can be removed beyond tree level. The effect of terms in the action that explicitly break chiral symmetry and therefore remove the phenomenon of species doubling are investigated by considering the energy-momentum relations of the arising tree-level improved actions. Our main result is that the O( a) improved action is a slightly modified Wilson fermion action can still be written with only nearest-neighbour fermion interactions.