Design of chalcogenide photovoltaic absorbers is carried out systematically through sequential cation mutation, from binary to ternary to quaternary compounds, using first-principles electronic structure calculations. Several universal trends are identified for two classes of quaternary chalcogenides (I2-II-IV-VI4 and I-III-II2-VI4 systems). For example, the lowest-energy structure always has larger lattice constant a, smaller tetragonal distortion parameter η = c/2a, and larger band gap than the metastable structures for common-row cation mutations. The band structure changes on mutation illustrate that although the band gap decreases from binary II-VI to ternary I-III-VI2 are mostly due to the p-d repulsion in the valence band, the decreases from ternary I-III-VI 2 to quaternary I2-II-IV-VI4 chalcogenides are due to the downshift in the conduction band caused by the wavefunction localization on the group IV cation site. We find that I2-II-IV- VI4 compounds are more stable in the kesterite structure, whereas the widely-assumed stannite structure reported in the literature is most likely due to partial disorder in the I-II (001) layer of the kesterite phase. Ten compounds are predicted have band gaps close to the 1 to 2 eV energy window suitable for photovoltaics.