Lattice QCD calculations with background magnetic fields are used to determine the magnetic moments of the octet baryons. Computations are performed at the physical value of the strange quark mass, and two values of the light quark mass, one corresponding to the S U ( 3 ) F -symmetric point, where the pion mass is m π ∼ 800     MeV , and the other corresponding to a pion mass of m π ∼ 450     MeV . The moments are found to exhibit only mild pion-mass dependence when expressed in terms of appropriately chosen magneton units the natural baryon magneton. A curious pattern is revealed among the anomalous baryon magnetic moments which is linked to the constituent quark model, however, careful scrutiny exposes additional features. Relations expected to hold in the large- N c limit of QCD are studied; and, in one case, a clear preference for the quark model over the large- N c prediction is found. The magnetically coupled Λ - Σ 0 system is treated in detail at the S U ( 3 ) F point, with the lattice QCD results comparing favorably with predictions based on S U ( 3 ) F symmetry. This analysis enables the first extraction of the isovector transition magnetic polarizability. The possibility that large magnetic fields stabilize strange matter is explored, but such a scenario is found to be unlikely.