Quorum systems have been used to implement many coordination problems in distributed systems. In this paper, we propose a reformulation of the definition of Byzantine quorum systems. Our reformulation captures the requirement for non-blocking access to quorums in asynchronous systems. We formally define the asynchronous access cost of quorum systems and we show that the asynchronous access cost and not the size of a quorum is the right measure of message complexity of protocols using quorums in asynchronous systems. We also show that previous quorum systems proposed in the literature have a very high asynchronous access cost. We present new quorum systems with low asynchronous access cost and whose other performance parameters match those of the best Byzantine quorum systems proposed in the literature. In particular, we present a construction for the disjoint failure pattern that outperforms previously proposed systems for that pattern.