  We develop a theory for a qualitatively new type of disorder in condensed
matter systems arising from local twist-angle fluctuations in two strongly
coupled van der Waals monolayers twisted with respect to each other to create a
flat band moir\'e superlattice. The new paradigm of 'twist angle disorder'
arises from the currently ongoing intense research activity in the physics of
twisted bilayer graphene. In experimental samples of pristine twisted bilayer
graphene, which are nominally free of impurities and defects, the main source
of disorder is believed to arise from the unavoidable and uncontrollable
non-uniformity of the twist angle across the sample. To address this new
physics of twist-angle disorder, we develop a real-space, microscopic model of
twisted bilayer graphene where the angle enters as a free parameter. In
particular, we focus on the size of single-particle energy gaps separating the
miniband from the rest of the spectrum, the Van Hove peaks, the renormalized
Dirac cone velocity near charge neutrality, and the minibandwidth. We find that
the energy gaps and minibandwidth are strongly affected by disorder while the
renormalized velocity remains virtually unchanged. We discuss the implications
of our results for the ongoing experiments on twisted bilayer graphene. Our
theory is readily generalized to future studies of twist angle disorder effects
on all electronic properties of moir\'e superlattices created by twisting two
coupled van der Waals materials with respect to each other.
