  In this paper, we extend the Atiyah--Guillemin--Sternberg convexity theorem
and Delzant's classification of symplectic toric manifolds to presymplectic
manifolds. We also define and study the Morita equivalence of presymplectic
toric manifolds and of their corresponding framed momentum polytopes, which may
be rational or non-rational. Toric orbifolds, quasifolds and non-commutative
toric varieties may be viewed as the quotient of our presymplectic toric
manifolds by the kernel isotropy foliation of the presymplectic form.
