This interesting paper is devoted to the study of bifurcation and dynamics of spiral waves. The transitions from meandering spiral waves to more complicated patterns can be observed in several chemical systems such as the Belousov-Zhabotinsky reaction or the catalysis on platinum surfaces. Chemical systems are traditionally modeled by reaction-diffusion systems \(u_t=d\triangle u+F(u,\mu)\) on suitable domains (\(u: \mathbb{R}^N \to \mathbb{R}^M\), \(\mu \) is a real parameter, \(d\) is a matrix with nonnegative entries, \(F\) is a smooth nonlinearity). Here the main modeling assumption is that \(x\in \mathbb{R}^N\), where \(N=2\) or \(N=3\). This system is equivariant with respect to the Euclidean symmetry group SE\((N)\), which is defined by the semidirect product \(\text{SO}(N)\dot +\mathbb{R}^N\). This group is the key to the dynamics of spiral waves. Here it is shown that the dynamics near meandering spiral waves or other patterns is determined by a finite-dimensional vector field that has a certain skew-product structure over the group SE\((N)\). The authors develop a systematic and rigorous procedure by which the equations-of-motion near relative periodic orbits can be derived. These results are formulated in an abstract functional-analytic set-up that includes the above stated reaction-diffusion equations equivariant under arbitrary finite-dimensional, and possibly noncompact, Lie groups. Hopf bifurcations of patterns occurring in reaction-diffusion equations in two and three dimensions are investigated. Some open problems are discussed as well.