The authors give a classification of Willmore canal surfaces, that is, of surfaces that envelope a \(1\)-parameter family of spheres while extremizing the Willmore functional \(\int(H^2-K) dA\). As a key step in their classification the authors show that a Willmore canal surface is isothermic: as a consequence any Willmore canal surface has to be Möbius equivalent to (a part of) a surface of revolution, a cylinder, or a cone in Euclidean space by a theorem of \textit{E.~Vessiot} [Bull. Soc. Math. Fr. 54, 139-176 (1924; JFM 52.0764.01) and 55, 39-79 (1925; JFM 53.0700.04)]. The generating curve of the surface is a free elastic curve in a hyperbolic \(2\)-plane, Euclidean plane, or the \(2\)-sphere, respectively. In the second half of the paper (sections 5 to 7) the authors derive explicit formulae for Willmore canal surfaces in terms of elliptic functions, [\textit{cf. J. Langer} and \textit{D. Singer}, J. Differ. Geom. 20, 1-22 (1984; Zbl 0554.53013)]. In order to derive their results the authors use the method of moving (Möbius) frames: the relevant background material on the conformal geometry of surfaces and, in particular, canal surfaces is elaborated in Sections 2 and 3 of the paper. The circle of problems around the classification given in the paper under review had caught the interest of other researchers: the fact that Willmore canal surfaces are Möbius equivalent to (a part of) a surface of revolution, a cylinder, or a cone, was independently obtained by \textit{K.~Voss} [Canal Willmore surfaces, Math. Forsch. Inst. Oberwolfach Tagungsbericht 45/1991], see also [\textit{P.~Tapernoux}, Willmore-Kanalflächen, Diploma Thesis, ETH Zürich (1991)]; as a consequence Willmore canal surfaces are Möbius-equivariant surfaces as discussed in \textit{D.~Ferus} and \textit{F.~Pedit} [Math. Z. 204, 269-282 (1990; Zbl 0722.53056), see also \textit{M.~Heil}, Über äquivariante Willmoreflächen in \(\mathbb{R}^3\); Diploma thesis, TU Berlin 1991]. It may also be mentioned that isothermic Willmore surfaces are, by \textit{G. Thomsen}'s theorem [Hamb. Math. Abh. 3, 31-56 (1923; JFM 49.0530.02)] or [\textit{W.~Blaschke}, Vorlesungen über Differentialgeometrie III, Springer Grundlehren Band 29, Berlin (1929; JFM 55.0422.01)], Möbius equivalent to minimal surfaces in space forms. In the case of Willmore canal surfaces these turn out to be equivariant in the ambient space form (this result is due to discussions with \textit{K.~Voss} and will be presented in greater detail in the reviewer's book [``Introduction to Möbius differential geometry'', to appear in the London Math. Soc. Lect. Note Ser.]), and the curvature of the corresponding ambient space form enters into the (elliptic) differential equation for the curvature of the meridian curve. In the reviewer's opinion there should be a relation between the authors' type classification of Willmore canal surfaces and the curvature of the ambient space of the surface as a minimal surface.