  We show how classical and quantum dualities, as well as duality relations
that appear only in a sector of certain theories ("emergent dualities"), can be
unveiled, and systematically established. Our method relies on the use of
morphisms of the "bond algebra" of a quantum Hamiltonian. Dualities are
characterized as unitary mappings implementing such morphisms, whose even
powers become symmetries of the quantum problem. Dual variables -which were
guessed in the past- can be derived in our formalism. We obtain new
self-dualities for four-dimensional Abelian gauge field theories.
