  We establish a general criterion for the finite presentability of subdirect
products of groups and use this to characterize finitely presented residually
free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group
is of type ${\rm{FP}}_n$ if and only if it is of type ${\rm{F}}_n$.
  New families of subdirect products of free groups are constructed, including
the first examples of finitely presented subgroups that are neither
${\rm{FP}}_\infty$ nor of Stallings-Bieri type. The template for these examples
leads to a more constructive characterization of finitely presented residually
free groups up to commensurability.
  We show that the class of finitely presented residually free groups is
recursively enumerable and present a reduction of the isomorphism problem. A
new algorithm is described which, given a finite presentation of a residually
free group, constructs a canonical embedding into a direct product of finitely
many limit groups.
  The (multiple) conjugacy and membership problems for finitely presented
subgroups of residually free groups are solved.
