Among all convex bodies, sections of ellipsoids and bodies of revolution exhibit particular symmetry. Namely, all hyperplanar sections of an ellipsoid are centrally symmetric and have an axis of symmetry, whereas all hyperplanar sections of a body of revolution have an axis of revolution. H. Brunn proved in 1889 that the central symmetry of all the sections characterizes ellipsoids. Much later, in 1965, C. A. Rogers observed that it is enough to consider only sections passing through a fixed point. Regarding axial symmetry, in 1999, K. Bezdek posed his celebrated conjecture that the axial symmetry of all the sections characterizes bodies of revolution in 3-dimensional space. Now, it is natural to formulate higher-dimensional analogs of Bezdek's conjecture, and there are many ways to do it. Our main result is a variant of Bezdek's conjecture in arbitrary dimension n≥3, where we assume that all the sections passing through a fixed point have an axis of symmetry, satisfying certain alignment condition. Further, if we weaken the hypothesis and consider only a 1-codimensional family of hyperplanes, we obtain a similar characterization of axially symmetric bodies. For each of these problems, we show both the orthogonal and the affine variant. Interestingly, in 3-dimensional space the proof is essentially different and touches on the theory of floating bodies. The talk is based on a joint work in progress with M. Angeles Alfonseca.