  A celebrated result by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich
shows that a set of integers is listable if and only if it is positive
existentially definable in the language of arithmetic. We investigate analogues
of this result over structures endowed with a listable presentation. When such
an analogue holds, the structure is said to have the DPRM property. We prove
several results addressing foundational aspects around this problem, such as
uniqueness of the listable presentation, transference of the DPRM property
under interpretation, and its relation with positive existential
bi-interpretability. A first application of our results is the rigorous proof
of (strong versions of) several folklore facts regarding transference of the
DPRM property. Another application of the theory we develop is that it will
allow us to link various Diophantine conjectures to the question of whether the
DPRM property holds for global fields. This last topic includes a study of the
number of existential quantifiers needed to define a Diophantine set.
