  Flavor singlet combinations of quark operators ${\cal{O}}_S^{\Gamma} =
\bar{u}\Gamma u + \bar{d}\Gamma d + \bar{s}\Gamma s$ contribute to many
important physical observables in the low energy region of QCD. Experimentally
one finds the values of some of these observables to be in sharp contrast to
the naive (perturbative) theoretical expectations. This indicates that non
perturbative vacuum properties might play a major role in the comprehension of
these phenomena. An example of such a vacuum contribution is the axial anomaly,
which appears in the divergence of the flavor singlet axial current and which
is connected to the topological properties of QCD. From a field theoretical
point of view flavor singlet matrix elements differ from non singlet amplitudes
in the occurrence of so called disconnected insertions. These are correlations
of hadron propagators with quark-antiquark loops or correlations between
quark-antiquark loops, which are mediated by vacuum fluctuations. According to
their respective flavor composition, the disconnected insertions cancel largely
in non singlet processes, but add in flavor singlet amplitudes. The lattice
approach provides an ideal tool to study flavor singlet phenomena. Being a
first principle method it should be capable to uncover non perturbative vacuum
contributions and to yield, on the long run, reliable results for the size of
such contributions in QCD. The present article reviews the status of flavor
singlet matrix element calculations in lattice QCD with respect to methods,
results and reliability. Special emphasis is paid to the discussion of state of
the art calculations of the pion nucleon sigma term $\sigma_{\pi N}$, the
flavor singlet axial coupling of the proton $G_A^1$, and the $\eta'$ mass.
