In this talk I will describe recent work on the snaking of spatially localized oscillations in two systems of interest, the cubic-quintic complex Swift-Hohenberg equation and a three-species reaction-diffusion system (the Purwins system) admitting a primary wave bifurcation, i.e. a Hopf bifurcation with a finite wavenumber. On a one-dimensional periodic domain this bifurcation gives rise simultaneously to two solution branches, standing and traveling waves of uniform amplitude, and we focus on the case where both bifurcate subcritically. We study in detail the properties, including stability, of spatially localized states (traveling pulses and spatial localized standing waves) originating in secondary bifurcations from these primary states. Tertiary bifurcations leading to drifting localized oscillations are also studied. These results are extended to wall-attached structures in a disk. These are found to mimic qualitatively the behavior in one spatial dimension, modified for some parameter values by the excitation of bulk modes of the disk. Our results are complemented with direct numerical simulations and compared with related results for breathers. This work is joint work with S. Modai, H. Uecker, N. Verschueren and A. Yochelis.