  We elaborate on the recent idea of a direct decomposition of Feynman
integrals onto a basis of master integrals on maximal cuts using intersection
numbers. We begin by showing an application of the method to the derivation of
contiguity relations for special functions, such as the Euler beta function,
the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function.
Then, we apply the new method to decompose Feynman integrals whose maximal cuts
admit 1-form integral representations, including examples that have from two to
an arbitrary number of loops, and/or from zero to an arbitrary number of legs.
Direct constructions of differential equations and dimensional recurrence
relations for Feynman integrals are also discussed. We present two novel
approaches to decomposition-by-intersections in cases where the maximal cuts
admit a 2-form integral representation, with a view towards the extension of
the formalism to $n$-form representations. The decomposition formulae computed
through the use of intersection numbers are directly verified to agree with the
ones obtained using integration-by-parts identities.
