  We propose definitions of regular and exact (virtual) double categories,
proving a number of results which parallel many basic results in the theory of
regular and exact categories. We show that any regular virtual double category
admits a factorization system which generalizes the factorization of a functor
between categories into a bijective-on-objects functor followed by a
fully-faithful functor. Finally, we show that our definition of exact double
category is equivalent to an axiom proposed by Wood, and very closely related
to the "tight Kleisli objects" studied by Garner and Shulman.
