"Constant Craving" peaked at number eight on Canada's RPM 100 Hit Tracks chart and number 38 on the US Billboard Hot 100, giving lang her only US top-40 hit. It also reached number two on the BillboardHot Adult Contemporary Tracks chart. In the United Kingdom, the song was a modest hit when released in 1992, but following a reissue in 1993 in the wake of its American success, it reached a new peak of number 15 on the UK Singles Chart.
John T. Davis from Austin American Statesman complimented "Constant Craving" as "a song graced by a compelling melody and a vocal brimming with yearning."[6]Larry Flick from Billboard stated, "There is no denying the rootsy, down-home quality of this wonderfully produced tune. Lang's distinctive voice is woven into spiraling multitracked harmonies, wafting over a base of acoustic guitar strumming and subtle accordian [sic] and xylophone interludes."[7] Steve Morse from Boston Globe named it the best song of the album, declaring it as "a romantic number that's also the only rhythmic tune of the bunch."[8] Clark and DeVaney from Cash Box named it Pick of the Week and a "standout" of the album.[9] A reviewer from Daily Mirror described it as "haunting".[10] Rufer and Fell from the Gavin Report commented, "k.d.'s foresaken country for a straight pop format that will finally win her the mass audience she's deserved. This track showcases her wonderful voice in solo and multi-part harmonies."[11]
Pan-European magazine Music & Media described it as "a Brecht/Weil on the prairie type of song—is likely to be lang's first hit in the UK and in Ireland".[12] On the album review, an Music & Media editor wrote, "Tales from the new west. La lang has changed her position on horseback for a more comfortable seat on the coach. In mind, she's singing in some Paris café in the decadent '20s."[13] John Milward from Rolling Stone felt the song is "more elusive", "but there's no denying the lift of its propulsive rhythms and joyful harmonies."[14] David Bauder from The Salt Lake Tribune praised "Constant Craving" as one of lang's best songs, remarking that it "concludes that a sense of longing is an inevitable part of life."[15]
Mark Deming from AllMusic felt that the "emotional core" of songs like "Constant Craving", "was obvious even when their surfaces were evasive."[16] In 2019, Billboard included it in its list of the "30 Lesbian Love Songs".[17] In 2008, the Daily Vault's Jason Warburg viewed it as a "dynamite single", that "manages to meld the spiritual with the physical while conveying a feverish desire for both kinds of fulfillment." He also remarked its "upbeat", "expansive and full of simmering appeal".[18]
The accompanying music video for the song was directed by American filmmaker and photographer Mark Romanek and filmed in black-and-white. It presents a fanciful recreation of the premiere of Samuel Beckett's play Waiting for Godot in Paris, 1953. Here, she is depicted singing backstage while the actors perform. The director says the song's lyrics of desperation and waiting fit well with the themes of Beckett's play.[19]VH1 added the video to its playlist in mid-April 1992.[20] The clip won Best Female Video at the 1993 MTV Video Music Awards.[21]
^Christensen, Thor (1998). "k.d. Lang". In Graff, Gary; Durchholz, Daniel (eds.). MusicHound Rock: The Essential Album Guide. Detroit: Visible Ink Press. pp. 657–658.
^Constant Craving (US cassette single sleeve). k.d. lang. Sire Records, Warner Bros. Records. 1992. 4-18942.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK 12-inch single sleeve). k.d. lang. Sire Records, Warner Bros. Records. 1992. W0100T, 9362-404550-2.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK CD single liner notes). k.d. lang. Sire Records, Warner Bros. Records. 1992. W0100CD, 9362-40455-2.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK 7-inch single sleeve). k.d. lang. Sire Records, Warner Bros. Records. 1992. W0100, 5439-18884-7.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK cassette single sleeve). k.d. lang. Sire Records, Warner Bros. Records. 1992. W0100C, 5439-18884-4.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (European CD single liner notes). k.d. lang. Sire Records, Warner Bros. Records. 1992. W0087CD, 9362-40432-2.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK 7-inch single vinyl disc). k.d. lang. Sire Records, Warner Bros. Records. 1993. W 0157, 5439-18589-7.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK cassette single sleeve). k.d. lang. Sire Records, Warner Bros. Records. 1993. W0157C, 5439-1859-4.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK CD1 & Australian CD single liner notes). k.d. lang. Sire Records, Warner Bros. Records. 1993. W0157 CD, 9362-40773-2.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)
^Constant Craving (UK CD2 liner notes). k.d. lang. Sire Records, Warner Bros. Records. 1993. W0157 CDX, 9362-40774-2.{{cite AV media notes}}: CS1 maint: others in cite AV media (notes) (link)