We consider the problem of covert communication over a state-dependent channel when the channel state is available either non-causally, causally, or strictly causally, either at the transmitter alone or at both transmitter and receiver. Covert communication with respect to an adversary, called "warden," is one in which, despite communication over the channel, the warden's observation remains indistinguishable from an output induced by innocent channel-input symbols. Covert communication involves fooling an adversary in part by a proliferation of codebooks; for reliable decoding at the legitimate receiver, the codebook uncertainty is typically removed via a shared secret key that is unavailable to the warden. In contrast to previous work, we do not assume the availability of a shared key at the transmitter and legitimate receiver. Instead, shared randomness is extracted from the channel state in a manner that keeps it secret from the warden, despite the influence of the channel state on the warden's output. When channel state is available at the transmitter and receiver, we derive the covert capacity region. When channel state is only available at the transmitter, we derive inner and outer bounds on the covert capacity. We provide examples for which the covert capacity is positive with knowledge of channel state information but is zero without it.