The objective of the project is to study jet cross sections in deeply inelastic lepton photon scattering in next-to-leading order QCD perturbation theory. The processes to be con- sidered on the Born level are those related to the production of one or two jets in the current fragmentation region and possibly one jet in the photon remnant fragmentation region for processes with a "resolved" photon. QCD corrections to be taken into account consist of virtual corrections and diagrams describing the emission of partons in the initial and final state of the QCD subprocess. The analytical results of the calculation will be implemented in a program suitable for the use by experimental physicists in order to carry out a data analysis. The results will be cast in a form that allows for the application of arbitrary jet definition schemes as well as experimental acceptance cuts. The theoretical results from the proposed project can be applied at high_energy e+e- _colliders like LEP and in particular LEP 2.