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 [16], quasifolds [3] and non-commutative toric varieties [14] may be viewed as the quotient of our presymplectic toric manifolds by the kernel isotropy foliation of the presymplectic form.