  Relying on the combinatorial classification of toric ideals using their
bouquet structure, we focus on toric ideals of hypergraphs and study how they
relate to general toric ideals. We show that hypergraphs exhibit a surprisingly
general behavior: the toric ideal associated to any general matrix can be
encoded by that of a $0/1$ matrix, while preserving the essential combinatorics
of the original ideal. We provide two universality results about the
unboundedness of degrees of various generating sets: minimal, Graver, universal
Gr\"obner bases, and indispensable binomials. Finally, we provide a
polarization-type operation for arbitrary positively graded toric ideals, which
preserves all the combinatorial signatures and the homological properties of
the original toric ideal.
