We analyze the potential of the CERN Large Hadron Collider (LHC) to study anomalous quartic vector-boson interactions Z Z gamma gamma, Z Z Z gamma, W+ W- gamma gamma, and W+ W- Z gamma through the weak boson fusion processes q q -> q q gamma gamma and q q -> q q gamma Z(-> l+ l-) with l = electron or muon. After a careful study of the backgrounds and how to extract them from the data, we show that the process p p -> j j gamma l+ l- is potentially the most sensitive to deviations from the Standard Model, improving the sensitivity to anomalous couplings by up to a factor 10^4 (10^2) with respect to the present direct (indirect) limits.