Short­cut graphs are graphs in which long enough cy­cles can­not embed with­out met­ric dis­tor­tion. Short­cut groups are groups which act prop­erly and co­com­pactly on short­cut graphs. These no­tions unify a sur­pris­ingly broad fam­ily of graphs and groups of in­ter­est in geo­met­ric group the­ory and met­ric graph the­ory in­clud­ing: sys­tolic and quadric groups (in par­tic­u­lar fi­nitely pre­sented C(6) and C(4)-T(4) small can­cel­la­tion groups), co­com­pactly cubu­lated groups, hy­per­bolic groups, Cox­eter groups and the Baum­slag-Soli­tar group BS(1,2). Most of these ex­am­ples sat­isfy a strong form of the short­cut prop­erty. I will dis­cuss some of these ex­am­ples as well as some gen­eral con­struc­tions and prop­er­ties of short­cut graphs and groups.