- In a one-pion plus one-kaon exchange model, we calculate the kaon production cross sections in nucleon-nucleon, nucleon-delta and delta-delta interactions. We find that this model describes reasonably well the experimental data on kaon production in the proton-proton interaction. Near the kaon production threshold, the cross section obtained from this model is smaller than that from the linear parametrization of Randrup and Ko. For kaon production cross sections from the nucleon-delta and delta-delta interactions, the cross sections are singular in free space, so we calculate them in a nuclear medium by including the (complex) pion self-energy. The results are compared with the scaling ansatz of Randrup and Ko. The theoretical cross sections are then used in a transport model to study kaon production from Au+Au collisions at 1 GeV/nucleon. 1995.