We consider random walks on comb- and brush-like graphs consisting of a base (of fractal dimension $D$) decorated with attached side-groups (branches). Random walks on comb-like structures were first studied as a simplified model of transport on percolation clusters, but in recent years the problem was also addressed from different perspectives and found to be relevant to many other natural and man-made systems and processes. We characterize branched graphs by the fractal dimension $D_a$ of a set of anchor points where side-groups are attached to the base. Two types of graphs are considered. Graphs of the first type are uniform in a sense that anchor points are distributed periodically over the base and thus form a subset of the base with dimension $D_a=D$. Graphs of the second type are decorated with side-groups in a regular yet non-uniform way: the set of anchor points has fractal dimension smaller than that of the base, $D_a<D$. For uniform graphs, a qualitative method to evaluate the sub-diffusion exponent suggested by Forte et al. for combs ($D=1$) is extended for brushes ($D>1$) and numerically tested for the Sierpinski brush (with the base and anchor set built on the same Sierpinski gasket). As an example of nonuniform graphs we consider the Cantor comb composed of one-dimensional base and side-groups, the latter attached to the former at anchor points forming the Cantor set. A peculiar feature of this and other nonuniform branched systems is a long-lived regime of super-diffusive transport when side-groups are of a finite size.