This article deals with a study of novel classes of metamaterial inclusions based on space-filling curves. The graph-theoretic Hamiltonian-path (HP) concept is exploited to construct a fairly broad class of space-filling curve geometries that include as special cases the well-known Hilbert and Peano curves whose application to metamaterial inclusions has recently been proposed. In this framework, the basic properties of HP are briefly reviewed, and a full-wave study of the electromagnetic properties of representative grid-graph HP geometries is carried out. Applications to metamaterial inclusions are explored, with focus on artificial magnetic conductors with reduced polarization-sensitivity.