c. 1300, curraunt, "running, flowing, moving along" (a sense now archaic), from Old French corant "running, lively, eager, swift," present participle of corre "to run," from Latin currere "to run, move quickly" (of persons or things), from PIE root *kers- "to run." Related: Currentness.
Sense of "presently in effect" is from mid-15c. Meaning "prevalent, generally reported or known" is from 1560s; that of "established by common consent" is from 1590s; that of "now passing, present now, in progress" is from c. 1600. Of money, "passing from one person to another," late 15c. Current events is attested from 1795; current affairs by 1776.