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Abstract Vortex Rossby waves (VRWs) affect tropical cyclones’ (TCs’) motion, structure, and intensity. They propagate within annular waveguides defined by a passband between Ω_1D, the Doppler-shifted frequency of a one-dimensional VRW, and zero. Wavenumber-1 VRWs cause TC motion directly and have wider waveguides than wavenumbers ≥ 2. VRWs forced with fixed rotation frequency propagate away from the forcing. Initially outward-propagating waves are Doppler shifted to zero at a critical radius, where they are absorbed. Initially inward-propagating waves are Doppler shifted to Ω_1D, reflect from a turning point, propagate outward, and are ultimately absorbed at the critical radius. Between the forcing and the turning radii, the VRWs have standing-wave structure; outward from the forcing they are trailing spirals. They carry angular momentum fluxes that act to accelerate the mean flow at the forcing radius and decelerate it at the critical radius. Mean-flow vorticity monopoles are inconsistent with Stokes’s theorem on a spherical Earth, because a contour enclosing the monopole’s antipode would have nonzero circulation but would enclose zero vorticity. The Rossby waveguide paradigm also fits synoptic-scale Rossby waves in a meridionally sheared zonal flow. These waves propagate within a waveguide confined between a poleward turning latitude and an equatorward critical latitude. Forced waves are comma-shaped gyres that resemble observed frontal cyclones, with trailing filaments equatorward of the forcing latitude and standing waves poleward. Even neutral forced Rossby waves converge westerly momentum at the latitude of forcing.