We continue our investigation of the unramified cohomology of smooth projective quadrics started in part I. Here we concentrate on quadrics X of dimension 2,3 and 4, and obtain detailed results. In particular, if X is defined by a "virtual Albert form" q, we relate the degree 4 unramified cohomology of X to the group PSO(q)/R, where R denotes Manin's R-equivalence.
Quadrics of dimension > 4 are studied in the forthcoming part III of this series.