Attempts to improve the efficiency of kesterite solar cells by changing the intrinsic stoichiometry have not helped to boost the device efficiency beyond the current record of 12.6%. In this light, the addition of extrinsic elements to the Cu2ZnSn(S,Se)4 matrix in various quantities has emerged as a popular topic aiming to ameliorate electronic properties of the solar cell absorbers. This article reviews extrinsic doping and alloying concepts for kesterite absorbers with the focus on those that do not alter the parent zinc-blende derived kesterite structure. The latest state-of-the-art of possible extrinsic elements is presented in the order of groups of the Periodic Table. The highest reported solar cell efficiencies for each extrinsic dopant are tabulated at the end. Several dopants like alkali elements and substitutional alloying with Ag, Cd or Ge have been shown to improve the device performance of kesterite solar cells as compared to the nominally undoped references, although it is often difficult to differentiate between pure electronic effects and other possible influences such as changes in the crystallization path, deviations in matrix composition and presence of alkali dopants coming from the substrates. The review is concluded with a suggestion to intensify efforts for identifying intrinsic defects that negatively affect electronic properties of the kesterite absorbers, and, if identified, to test extrinsic strategies that may compensate these defects. Characterization techniques must be developed and widely used to reliably access semiconductor absorber metrics such as the quasi-Fermi level splitting, defect concentration and their energetic position, and carrier lifetime in order to assist in search for effective doping/alloying strategies.