We introduce the notion of a continuous verifiable delay function (cVDF): a function g which is (a) iteratively sequential—meaning that evaluating the iteration g(t) of g (on a random input) takes time roughly t times the time to evaluate g, even with many parallel processors, and (b) (iteratively) verifiable—the output of g(t) can be efficiently verified (in time that is essentially independent of t). In other words, the iterated function g(t) is a verifiable delay function (VDF) (Boneh et al., CRYPTO ’18), having the property that intermediate steps of the computation (i.e., g (t') for t'< t) are publicly and continuously verifiable. We demonstrate that cVDFs have intriguing applications: (a) they can be used to construct public randomness beacon that only require an initial random seed (and no further unpredictable sources of randomness), (b) enable outsourceable where any part of the VDF computation can be verifiably outsourced, and (c) have deep complexity-theoretic consequences: in particular, they imply the existence of depth-robust moderately-hard Nash equilibrium problem instances, i.e. instances that can be solved in polynomial time yet require a high sequential running time. Our main result is the construction of a cVDF based on the repeated squaring assumption and the soundness of the Fiat-Shamir (FS) heuristic for constant-round proofs. We highlight that when viewed as a (plain) VDF, our construction requires a weaker FS assumption than previous ones (earlier constructions require the FS heuristic for either super-logarithmic round proofs, or for arguments).