Cylindric skew Schur functions, which are a generalization of skew Schur functions, arise naturally in the study of P-partitions. They are also closely related to the fundamental open problem of finding a combinatorial proof of the positivity of the 3-point Gromov-Witten invariants. After explaining these motivations, we ask when a cylindric skew Schur function is Schur-positive, i.e. has all positive coefficients when expanded in terms of Schur functions. Using a result of I. Gessel and C. Krattenthaler, we generalize a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving a user-friendly tool for expanding an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.