We analyse the all-pay auction with incomplete information and variance-averse bidders. We characterise the symmetric equilibrium for general distributions of valuations and any number of bidders. Variance aversion is a sufficient assumption to predict that high-valuation bidders increase their bids relative to the risk-neutral case while low types decrease their bid. Considering an asymmetric two-player environment with uniform valuations, we show that a more variance-averse type always bids higher than her less variance-averse counterpart. Utilising our analytical bidding functions we discuss all-pay auctions with variance-averse bidders from a designer's perspective. We extend our basic model to include noisy signals and allow for the possibility of variance-seeking preferences and type-dependent attitudes towards risk.