  We review recent studies for superconductivity using diagrammatic extensions
of dynamical mean field theory. These approaches take into account
simultaneously both, the local correlation effect and spatial long-range
fluctuations, which are essential to describe unconventional superconductivity
in a quasi-two-dimensional plane. The results reproduce and predict the
experimental phase diagrams of strongly correlated system such as cuprates and
nickelates. Further studies reveal that the dynamical screening effect of the
pairing interaction vertex has dramatic consequences for the transition
temperature and may even support exotic mechanisms like odd-frequency pairing.
We also discuss the dimensionality of layered materials and how to interpret
the numerical results in two dimensions.
