Abstract Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set X of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets $${{\,\mathrm{\texttt{C}}\,}}(v)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mi>C</mml:mi> <mml:mspace /> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of descendant taxa of a vertex v . The clustering system $$\mathscr {C}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> comprising the clusters of a network N conveys key information on N itself. In the special case of rooted phylogenetic trees, T is uniquely determined by its clustering system $$\mathscr {C}_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> . Although this is no longer true for networks in general, it is of interest to relate properties of N and $$\mathscr {C}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> . Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering systems of the following form: If N is a network of type $$\mathbb {X}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , then $$\mathscr {C}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> satisfies $$\mathbb {Y}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> , and conversely if $$\mathscr {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> is a clustering system satisfying $$\mathbb {Y},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> then there is network N of type $$\mathbb {X}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> such that $$\mathscr {C}\subseteq \mathscr {C}_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>⊆</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> .This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.