We introduce and study kernel algebras , i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as D-modules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of [44] concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon [33] concerning an analog of the Fourier transform for derived categories of quasi-coherent sheaves on a dual pair of generalized 1-motives.