We propose a class of novel variance-reduced stochastic conditional gradient methods. By adopting the recent stochastic path-integrated differential estimator technique (SPIDER) of Fang et al. (2018) for the classical Frank-Wolfe (FW) method, we introduce SPIDER-FW for finite-sum minimization as well as the more general expectation minimization problems. SPIDER-FW enjoys superior complexity guarantees in the non-convex setting while matching the best FW variants in the literature in the convex case. We also extend our framework `a la conditional gradient sliding (CGS) of Lan & Zhou (2016) and propose SPIDER-CGS to further reduce the stochastic first-order oracle complexity.