Old English crafian "ask, implore, demand by right," from North Germanic *krabojan (source also of Old Norse krefja "to demand," Danish kræve, Swedish kräva); perhaps related to craft (n.) in its base sense of "power." Current sense "to long for, eagerly desire" is c. 1400, probably through intermediate meaning "to ask very earnestly" (c. 1300). Related: Craved; craving.
"vehement or urgent desires," by 1711, from craving, verbal noun from crave.
c. 1200, cravant "defeated, vanquished, overcome, conquered," apparently adapted from Old French cravent "defeated, beaten," past participle of cravanter "to strike down, to fall down," from Latin crepare "to crack, creak" (see raven). The sense, apparently affected by crave, shifted from "defeated" to "cowardly" (c. 1400) perhaps via intermediary sense of "confess oneself defeated." As a noun, "an acknowledged coward," 1580s. Related: Cravenly; cravenness.
c. 1300, "inordinate desire of gaining and possessing wealth," fifth of the seven deadly sins, from Old French avarice "greed, covetousness" (12c.), from Latin avaritia "greed, inordinate desire," from avarus "greedy, grasping," adjectival form of avere "crave, long for, be eager," from Proto-Italic *awe- "to be eager," from PIE *heu-eh- "to enjoy, consume" (source also of Sanskrit avasa- "refreshment, food," avisya- "gluttony;" Welsh ewyllys "will;" Armenian aviwn "lust").
In Middle English it also was used of immoderate desire for knowledge, glory, power, etc., but it "has become limited, except in figurative uses, so as to express only a sordid and mastering desire to get wealth" [Century Dictionary].