We develop a theory for a 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 moire superlattice. The 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 nonuniformity of the twist angle across the sample. To address this 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 moire superlattices created by twisting two coupled van der Waals materials with respect to each other.