Definable coherent sheaves (with respect to an o-minimal structure) were introduced by Bakker, Brunebarbe and Tsimerman  (BBT) and used as an essential tool in their proof of Griffiths' conjecture that the image of the period map is algebraic.   The category of these definable sheaves on a complex algebraic variety X sits in between the category of algebraic and analytic sheaves.  More precisely, there is a definablization functor taking coherent algebraic sheaves to definable coherent sheaves and an analytification functor going from the category of definable coherent sheaves to the category of coherent analytic sheaves.  This makes them useful for answering questions about analytic maps involving algebraic varieties.  I'll explain these two functors and the concept of o-minimality necessary to define the BBT category of definable coherent sheaves.  Then I'll state a couple of results I obtained recently with Adam Melrod on the cohomology groups of definable coherent sheaves both in the case where X is projective (when, for reasonable o-minimal structures,  the groups are the same as the usual cohomology groups) and the general case (when they very much aren't).