An up-to-date text, covering the concept of incomplete fusion (ICF) in heavy ion (HI) interactions at energies below 10 MeV/ nucleon. Important concepts including the exciton model, the Harp Miller and Berne model, Hybrid model, Sum rule model, Hot spot model and promptly emitted particles model are covered in depth. It studies the ICF and PE-emission in heavy ion reactions at low energies using off-beam and in-beam experimental techniques. Theories of complete fusion (CF) of heavy ions based on Compound Nucleus (CN) mechanism of statistical nuclear reactions, details of the Computer code PACE4 based on CN mechanism, pre-equilibrium (PE) emission, modeling of (ICF) and their limits of application are discussed in detail.