  Within class of generic black holes there are extremal black holes (with
vanishing Hawking temperature T) and vanishing horizon area Ah, but with finite
Ah/T ratio,the Extremal Vanishing Horizon (EVH) black holes. We study the near
horizon limit of a four dimensional EVH black hole solution to a generic
(gauged) Einstein-Maxwell dilaton theory and show that in the near horizon
limit they develop a throat which is a pinching orbifold limit of AdS3. This is
an extension of the well known result for extremal black holes the near horizon
limit of which contains an AdS2 throat. We show that in the near EVH near
horizon limit the pinching AdS3 factor turns to a pinching BTZ black hole and
that this near horizon limit is indeed a decoupling limit. We argue that the
pinching AdS3 or BTZ orbifold is resolved if the near horizon limit is
accompanied by taking the 4d Newton constant G4 to zero such that the
Bekenstein-Hawking entropy S = Ah/(4G4) remains finite. We propose that in this
limit the near horizon EVH black hole is dual to a 2d CFT. We provide pieces of
evidence in support of the EVH/CFT correspondence and comment on its connection
to the Kerr/CFT proposal and speculations how the EVH/CFT may be used to study
generic e.g. Schwarzchild-type black holes.
