The main purpose of this paper is to introduce various convexity concepts in terms of a positive Chebyshev system ω and give a systematic investigation of the relations among them. We generalize a celebrated theorem of Bernstein–Doetsch to the setting of ω-Jensen convexity. We also give sufficient conditions for the existence of discontinuous ω-Jensen affine functions. The concept of Wright convexity is extended to the setting of Chebyshev systems, as well, and it turns out to be an intermediate convexity property between ω-convexity and ω-Jensen convexity. For certain Chebyshev systems, we generalize the decomposition theorems of Wright convex and higher-order Wright convex functions obtained by C. T. Ng in 1987 and by Maksa and Páles in 2009, respectively.