  We study the three-body Coulomb problem in two dimensions and show how to
calculate very accurately its quantum properties. The use of a convenient set
of coordinates makes it possible to write the Schr\"{o}dinger equation only
using annihilation and creation operators of four harmonic oscillators, coupled
by various terms of degree up to twelve. We analyse in details the discrete
symmetry properties of the eigenstates. The energy levels and eigenstates of
the two-dimensional helium atom are obtained numerically, by expanding the
Schr\"{o}dinger equation on a convenient basis set, that gives sparse banded
matrices, and thus opens up the way to accurate and efficient calculations. We
give some very accurate values of the energy levels of the first bound Rydberg
series. Using the complex coordinate method, we are also able to calculate
energies and widths of doubly excited states, i.e. resonances above the first
ionization threshold. For the two-dimensional $H^{-}$ ion, only one bound state
is found.
