In this work, we study the dynamical properties of Krystyna Kuperberg's aperiodic flows on 3-manifolds. We introduce the notion of a “zippered lamination,” and with suitable generic hypotheses, show that the unique minimal set for such a flow is an invariant zippered lamination. We obtain a precise description of the topological and dynamical properties of the minimal set, including the presence of non-zero entropy-type invariants and chaotic behavior. Moreover, we show that the minimal set does not have stable shape, yet satisfies the Mittag-Leffler condition for homology groups.