  We tackle the challenge of ensuring the deadlock-freedom property for
message-passing processes that communicate asynchronously in cyclic process
networks. Our contributions are twofold. First, we present Asynchronous
Priority-based Classical Processes (APCP), a session-typed process framework
that supports asynchronous communication, delegation, and recursion in cyclic
process networks. Building upon the Curry-Howard correspondences between linear
logic and session types, we establish essential meta-theoretical results for
APCP, most notably deadlock freedom. Second, we present a new concurrent
$\lambda$-calculus with asynchronous session types, dubbed LASTn. We illustrate
LASTn by example and establish its meta-theoretical results; in particular, we
show how to soundly transfer the deadlock-freedom guarantee from APCP. To this
end, we develop a translation of terms in LASTn into processes in APCP that
satisfies a strong formulation of operational correspondence.
