We consider the orbital magnetic properties of noninteracting charge carriers in graphene-based nanostructures in the low-energy regime. The magnetic response of such systems results both from bulk contributions and from confinement effects that can be particularly strong in ballistic quantum dots. First we provide a comprehensive study of the magnetic susceptibility chi of bulk graphene in a magnetic field for the different regimes arising from the relative magnitudes of the energy scales involved, i.e., temperature, Landau-level spacing, and chemical potential. We show that for finite temperature or chemical potential, chi is not divergent although the diamagnetic contribution chi(0) from the filled valance band exhibits the well-known -B-1/2 dependence. We further derive oscillatory modulations of chi, corresponding to de Haas-van Alphen oscillations of conventional two-dimensional electron gases. These oscillations can be large in graphene, thereby compensating the diamagnetic contribution chi(0) and yielding a net paramagnetic susceptibility for certain energy and magnetic field regimes. Second, we predict and analyze corresponding strong, confinement-induced susceptibility oscillations in graphene-based quantum dots with amplitudes distinctly exceeding the corresponding bulk susceptibility. Within a semiclassical approach we derive generic expressions for orbital magnetism of graphene quantum dots with regular classical dynamics. Graphene-specific features can be traced back to pseudospin interference along the underlying periodic orbits. We demonstrate the quality of the semiclassical approximation by comparison with quantum-mechanical results for two exemplary mesoscopic systems, a graphene disk with infinite mass-type edges, and a rectangular graphene structure with armchair and zigzag edges, using numerical tight-binding calculations in the latter case.