This thesis concerns the theoretical study of the low-energy interaction of antihydrogen atoms (¯H) with hydrogen molecules and antiprotons (p¯) with hydrogen molecular ions. Both systems are of interest to experiments currently under way at CERN aiming to trap and study antihydrogen atoms. To date limited work has been carried out on these systems. Only four papers on the ¯H-H2 system have been published and so far only semi-classical results have been published for p¯-H+ 2 . For both systems potential energy surfaces are constructed by fitting analytical functions to ab initio calculated energies. A combination of a neural network and suitable long-range asymptotic functions are used to fit the surfaces of both systems. The resulting surfaces are more accurate than those of previous work and should be useful for further studies. Quantum mechanical scattering calculations are carried out using the S-matrix Kohn variational method. It is shown that basis sets which are effective for normal chemical systems are not suitable for mixed matter/antimatter interactions. Instead, a basis set of multidimensional Gaussian functions, tailored to the potential, is used to converge the scattering calculations. For ¯H-H2 scattering the diatomic is approximated as a rigid rotor. Elastic and rotationally inelastic scattering calculations are presented. Hadronic annihilation calculations are also presented along with the first calculation of leptonic annihilation cross sections for this system. Scattering calculations for the p¯-H+ 2 system are carried out with approximations made to the potential. The strength of the interaction between the antiproton and hydrogen molecular ion, which at long range becomes Coulombic, requires large basis sets. The necessary modifications required to use the Kohn method with a Coulomb potential are discussed. A preliminary calculation of Pn-H elastic scattering is also carried out.