  Recent years have seen a surprising connection between the physics of
scattering amplitudes and a class of mathematical objects--the positive
Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which
have been loosely referred to as "positive geometries". The connection between
the geometry and physics is provided by a unique differential form canonically
determined by the property of having logarithmic singularities (only) on all
the boundaries of the space, with residues on each boundary given by the
canonical form on that boundary. In this paper we initiate an exploration of
"positive geometries" and "canonical forms" as objects of study in their own
right in a more general mathematical setting. We give a precise definition of
positive geometries and canonical forms, introduce general methods for finding
forms for more complicated positive geometries from simpler ones, and present
numerous examples of positive geometries in projective spaces, Grassmannians,
and toric, cluster and flag varieties. We also illustrate a number of
strategies for computing canonical forms which yield interesting
representations for the forms associated with wide classes of positive
geometries, ranging from the simplest Amplituhedra to new expressions for the
volume of arbitrary convex polytopes.
