  The need for accurate calculations on atoms and diatomic molecules is
motivated by the opportunities and challenges of such studies. The most
commonly-used approach for all-electron electronic structure calculations in
general - the linear combination of atomic orbitals (LCAO) method - is
discussed in combination with Gaussian, Slater a.k.a. exponential, and
numerical radial functions. Even though LCAO calculations have major benefits,
their shortcomings motivate the need for fully numerical approaches based on,
e.g. finite differences, finite elements, or the discrete variable
representation, which are also briefly introduced.
  Applications of fully numerical approaches for general molecules are briefly
reviewed, and their challenges are discussed. It is pointed out that the high
level of symmetry present in atoms and diatomic molecules can be exploited to
fashion more efficient fully numerical approaches for these special cases,
after which it is possible to routinely perform all-electron Hartree-Fock and
density functional calculations directly at the basis set limit on such
systems. Applications of fully numerical approaches to calculations on atoms as
well as diatomic molecules are reviewed. Finally, a summary and outlook is
given.
