We review the behavior of stationary and moving spatially periodic patterns in a simple cross-flow fixed-bed reactor with a first-order exothermic reaction subject to the Danckwert's boundary conditions and realistically high Le and Pe. Spatiotemporal patterns emerge due to the interaction of concentration and temperature balances, much like dynamic patterns in a CSTR. Moving waves emerge in an unbounded system, but they transform into stationary spatially inhomogeneous patterns in a bounded system above certain Pe threshold. The critical parameters of this threshold are derived analytically. A weakly nonlinear analysis is used in order to derive the governing amplitude equation. The spatial behavior in the bounded system with Pe --> infinity is analogous to the temporal behavior of the simple thermokinetic CSTR problem and the behavior of the distributed system is classified according to that of the lumped one. Both regular kinetics and oscillatory one (with reversible catalytic activity) are considered. Suggestions for experimental realization of these phenomena are discussed.