Ultraperipheral collisions (UPCs) at the LHC and RHIC provide important new information on the partonic structure of the proton and nuclei and small-$x$ dynamics in QCD. We review phenomenological applications of the collinear factorization at leading and next-to-leading orders of perturbative QCD and the dipole model to coherent and incoherent $J/\psi$ photoproduction in Pb-Pb UPCs at the LHC emphasizing the strong leading twist gluon nuclear shadowing, the role of quark-antiquark-gluon dipoles, and a possible onset the gluon saturation in nuclei. We also discuss inclusive and diffractive dijet photoproduction in UPCs, which give complementary constraints on nuclear parton distributions and the pattern of factorization breaking in diffraction.