The eigenvalue problem of a Hamiltonian describing two interacting particles confined by an external harmonic-oscillator potential is analyzed. If the particles are identical, the system is referred to as harmonium. Harmonium belongs to very few non-trivial two-particle systems for which the Schrödinger equation is not only separable but also, for several interaction potentials and for a set of the coupling constants, quasi-exactly solvable. In particular, the complete spectrum of harmonium has been obtained and analyzed. Also some aspects or relativistic generalizations of the model are briefly discussed.