Quasiconvex set-valued functions appear naturally as risk measures for random vectors in financial mathematics. Motivated by some examples in systemic risk, in this talk, we will consider some recent developments on quasiconvex monotone set-valued functions with a particular emphasis on duality theory. In applications, one usually needs to study the composition of two quasiconvex monotone set-valued functions, which is not a quasiconvex function in general. As a remedy, we will discuss the so-called natural quasiconvexity property and its consequences in the study of compositions.