  In this paper we give a complete characterization of those knotted toroidal
sets that can be realized as attractors for both discrete and continuous
dynamical systems globally defined in $\mathbb{R}^3$. We also see that the
techniques used to solve this problem can be used to give sufficient conditions
to ensure that a wide class of subcompacta of $\mathbb{R}^3$ that are
attractors for homeomorphisms must also be attractors for flows. In addition we
study certain attractor-repeller decompositions of $\mathbb{S}^3$ which arise
naturally when considering toroidal sets.
