Abstract S-wave and P-wave superconductivity in Kondo lattice is theoretically studied in connection with heavy-fermion superconductivity (HFS). The hybridization between f-electron and conduction electron is investigated consistently and the corresponding quantities have been calculated in detail in the generalized Nambu formalism. It is shown that if one thinks f-electrons are responsible for superconductivity, for S-wave paired state, the superconducting transition temperature is in agreement with that of Tachiki et al., but the specific heat jump is not, our result is more appropriate than theirs. For P-wave paired state, the HFSs described by a Kondo lattice model are essentially almost localized Fermi superfluids, with a small modification. The effects of impurities on S- and P-wave paired states have been studied in detail respectively, and the conditions of the appearence of gapless superconductivity have been obtained for each case .