We consider the secret sharing problem, in which a dealer distributes a secret among a set of participants in such a manner that only qualified sets of users can recover the secret by pooling their shares together while non-qualified sets of users will obtain no information about the secret even if they pool their shares together. In contrast to the existing solutions that are mainly based on number theoretic tools, we propose a physical layer approach that exploits the presence of random noise inherent to wireless channels for secret sharing. Two different scenarios are considered. In the first scenario, the classic secret sharing problem with a single secret message is considered, in which qualified sets are specified by a general access structure. A secret sharing scheme is proposed by constructing a secure coding scheme for an equivalent compound wiretap channel. Based on this approach, both lower and upper bounds on the secret sharing capacity are obtained. For some special cases, the secret sharing capacity is fully characterized. In the second scenario, a generalization of the classic secret sharing problem is proposed, in which multiple secret messages are required to be recovered at different qualified sets. A secret sharing scheme is provided by constructing an equivalent broadcast channel with compound eavesdroppers and constructing a secure coding scheme for the equivalent channel.