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Elemental abundances for volatile siderophile and chalcophile elements for Mars inform us about processes of accretion and core formation. Such data are few for Martian meteorites, and are often lacking in the growing number of desert finds. In this study, we employed laser ablation inductively coupled plasma-mass spectrometry (LA-ICP-MS) to analyze polished slabs of 15 Martian meteorites for the abundances of about 70 elements. This technique has high sensitivity, excellent precision, and is generally accurate as determined by comparisons of elements for which literature abundances are known. However, in some meteorites, the analyzed surface is not representative of the bulk composition due to the over- or underrepresentation of a key host mineral, e.g., phosphate for rare earth elements (REE). For other meteorites, the range of variation in bulk rastered analyses of REE is within the range of variation reported among bulk REE analyses in the literature. An unexpected benefit has been the determination of the abundances of Ir and Os with a precision and accuracy comparable to the isotope dilution technique. Overall, the speed and small sample consumption afforded by this technique makes it an important tool widely applicable to small or rare meteorites for which a polished sample was prepared. The new volatile siderophile and chalcophile element abundances have been employed to determine Ge and Sb abundances, and revise Zn, As, and Bi abundances for the Martian mantle. The new estimates of Martian mantle composition support core formation at intermediate pressures (14 +/- 3GPa) in a magma ocean on Mars.