  A few years ago various disparities for Laplacians on graphs and manifolds
were discovered. The corresponding results are mostly related to volume growth
in the context of unbounded geometry. Indeed, these disparities can now be
resolved by using so called intrinsic metrics instead of the combinatorial
graph distance. In this article we give an introduction to this topic and
survey recent results in this direction. Specifically, we cover topics such as
Liouville type theorems for harmonic functions, essential selfadjointness,
stochastic completeness and upper escape rates. Furthermore, we determine the
spectrum as a set via solutions, discuss upper and lower spectral bounds by
isoperimetric constants and volume growth and study $p$-independence of spectra
under a volume growth assumption.
