The solar chemical composition is an important ingredient in our understanding of the formation, structure, and evolution of both the Sun and our Solar System. Furthermore, it is an essential reference standard against which the elemental contents of other astronomical objects are compared. In this review, we evaluate the current understanding of the solar photospheric composition. In particular, we present a redetermination of the abundances of nearly all available elements, using a realistic new three-dimensional (3D), time-dependent hydrodynamical model of the solar atmosphere. We have carefully considered the atomic input data and selection of spectral lines, and accounted for departures from local thermodynamic equilibrium (LTE) whenever possible. The end result is a comprehensive and homogeneous compilation of the solar elemental abundances. Particularly noteworthy findings are significantly lower abundances of C, N, O, and Ne compared to the widely used values of a decade ago. The new solar chemical composition is supported by a high degree of internal consistency between available abundance indicators, and by agreement with values obtained in the Solar Neighborhood and from the most pristine meteorites. There is, however, a stark conflict with standard models of the solar interior according to helio-seismology, a discrepancy that has yet to find a satisfactory resolution.