The Knight shift, % , measures th e magne ti c hype rfin e fi eld at th e nucleu s produ ced by th e conducti on e lec tro ns whi c h are po la ri ze d in a ma gne ti c fi e ld. Kni ght s hifts are ofte n dominated by th e Pauli term a nd , in it.s mo st s impl e form , can be writte n as % = (a) x p. H ere XP is the conduction electron Pauli s pin s usce ptibility whic h de pe nd s on th e de nsity of states at th e Fe rmi le vel, N(EF), and (a ) is an a ve rage magneti c hype rfin e couplin g' cons tant associated with th e wav e function characte r at the nu c le us, 1 1,.( 0) I", for co ndu c ti o n ele ctrons at th e F e rmi surface. The Kni ght s hift th~refore provides, through ( a ), insight into th e wave-function c haracte r assoc iated wit.h N(E,.·). Ca lc ul a ti ons of ( a ) involvin g a n a ve raging ove r k-s pace have bee n a tte mpted for a fe w simple me tals up to the prese nt tim e. For alloys and inte rm etaJJi c co mpound s, rath er diffe re nt ( a ) 's are expe rim entall y obse rved for diffe re nt local environm e nts, indi ca tin g that % sampl es th e variati on in local wav e-fun cti o n c ha racte r, or a va ri ati on in local de nsit y of sta tes . Th e re is no uniqu e wa y of se para tin g th e local va ri ati on of N(E,.) fro m 1 1,· 1 (0) 12.In thi s article the me thods deve loped for relatin g % to t.h e e lectroni c prope rti es for most of th e types of cases e nco untered in the literature are re vi e wed. We di sc uss "simple" me tal s in cluding probl e ms of orbital magne tis m a nd cha nge s in % cau sed by electroni c tran siti ons suc h as meltin g. Kni ght s hifts and the ir te mpe ra ture de pe nd e nce in metals and inte rme taJJic compo und s involvin g unfill ed d s hells , are di sc ussed . We give es timates of atomi c hype rfin e fi eld s due to sin gle electrons, appropriate to those cases where proble ms du e to electronic configurations do not mak e deduc ti ons from expe rime nt too ambiguou s. A density of states c urve calc ulated for Cu is given , showing th e relative importance of Sop , a nd d c ha racte r for th a t me tal. In a qualitative sense this C u c urve implies such information for other transition metals. We disc uss alloy solid solutions for the cases where a "rigid" band mod el might be us ed to explain the res ults, and for cases where local effects have to be taken into accounl. The charge oscillation and RKKY approac hes and their limitations are reviewed for cases of dilute nonm agne ti c a nd d-or !-t.ype impurities.Key word s: El ectroni c densit.y of states; hyperfin e fields; Knight shift; nuclear magnetic reso nan ce; s usce ptibility; wa ve fun cti ons.