This thesis describes the calculations of masses and matrix elements using lattice QCD. Exploratory results are presented for the hadron mass spectrum and pseudoscalar meson decay constants using a mixed action formalism where the sea and valence actions are different. Improved staggered sea quarks are used and HYP-smeared overlap valence quarks. A method for matching the sea and valence quark masses is proposed. Good signals on 10 configurations at one lattice spacing and two different sets of sea quark masses are obtained. Results are presented for light meson masses and pseudoscalar meson decay constants in 2+1 flavour domain wall QCD with the DBW2 and Iwasaki gauge actions. This formalism preserves chiral symmetry at finite lattice spacing. The lattices used have linear sizes in the range 1.6 to 2.2fm and u and d quark masses as low as one quarter of the strange quark mass. All data were generated on the QCDOC machines at the University of Edinburgh and Brookhaven National Laboratory. Despite large residual masses and a limited number of sea quark mass values with which to perform chiral extrapolations, our results agree with experiment and scale within errors.