  In PRL 116 (2016) no.6, 062001, the space of planar pentagon functions that
describes all two-loop on-shell five-particle scattering amplitudes was
introduced. In the present paper we present a natural extension of this space
to non-planar pentagon functions. This provides the basis for our pentagon
bootstrap program. We classify the relevant functions up to weight four, which
is relevant for two-loop scattering amplitudes. We constrain the first entry of
the symbol of the functions using information on branch cuts. Drawing on an
analogy from the planar case, we introduce a conjectural second-entry condition
on the symbol. We then show that the information on the function space, when
complemented with some additional insights, can be used to efficiently
bootstrap individual Feynman integrals. The extra information is read off of
Mellin-Barnes representations of the integrals, either by evaluating simple
asymptotic limits, or by taking discontinuities in the kinematic variables. We
use this method to evaluate the symbols of two non-trivial non-planar
five-particle integrals, up to and including the finite part.
