We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family P of possible physical measures. A robust notion NA1(P) of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: NA1(P) holds if and only if every P ∈ P admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.