We introduce here the notion of exhaustiveness, which is related with the notion of equicontinuity, in asymmetric metric spaces. We give the relation between equicontinuity and exhaustiveness in such spaces and some theorems and results about it. We show that in the asymmetric situation forward convergence does not imply backward convergence (or vice versa), the limit of a sequence of exhaustive functions may not be continuous, also may not be unique. Also, we prove a type of Ascoli theorem using the notion of exhaustiveness in the asymmetric case. Finally, following Caserta and Kocinac , we will investigate some properties of a statistical version of exhaustiveness in asymmetric metric spaces.