  We define the notion of a twisted topological graph algebra associated to a
topological graph and a $1$-cocycle on its edge set. We prove a stronger
version of a Vasselli's result. We expand Katsura's results to study twisted
topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness
theorem, describe the gauge-invariant ideal structure. We find that a twisted
topological graph algebra is simple if and only if the corresponding untwisted
one is simple.
