  This thesis aims to develop a compositional theory for the operational
semantics of networks. The networks considered are described by either internal
or enriched graphs. In the internal case we focus on $\mathsf{Q}$-nets, a
generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$.
$\mathsf{Q}$-nets include many known variants of Petri nets including pre-nets,
integer nets, elementary net systems, and bounded nets. In the enriched case we
focus on graphs enriched in a quantale $R$ regarded as matrices with entries in
$R$. These $R$-matrices represent distance networks, Markov processes, capacity
networks, non-deterministic finite automata, simple graphs, and more. The
operational semantics of $\mathsf{Q}$-nets is constructed as an adjunction
between $\mathsf{Q}$-nets and categories internal to the category of models of
$\mathsf{Q}$. Similarly, the operational semantics of $R$-matrices is
constructed as an adjunction between $R$-matrices and categories enriched in
$R$. The left adjoint of this adjunction sends an $R$-matrix $M$ to the
$R$-category $F_R(M)$ whose hom-objects are solutions of the algebraic path
problem: a generalization of the shortest path problem to graphs weighted in
$R$. For both $\mathsf{Q}$-nets and $R$-matrices we use the theory of
structured cospans to study the compositionality of the above operational
semantics. For each type of network we construct a double category whose
morphisms are "open networks", i.e. networks with certain vertices designated
as input or output. We introduce the black-boxing of an open network, a
profunctor describing the externally observable behavior of an open network. We
introduce a class of open networks called "functional open networks" for which
black-boxing preserves composition.
