  Topological Ramsey spaces are spaces which support infinite dimensional
Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space
is endowed with a partial ordering which can be modified to a $\sigma$-closed
`almost reduction' relation analogously to the partial ordering of `mod finite'
on $[\omega]^{\omega}$. Such forcings add new ultrafilters satisfying weak
partition relations and have complete combinatorics. In cases where a forcing
turned out to be equivalent to a topological Ramsey space, the strong
Ramsey-theoretic techniques have aided in a fine-tuned analysis of the
Rudin-Keisler and Tukey structures associated with the forced ultrafilter and
in discovering new ultrafilters with complete combinatorics.This expository
paper provides an overview of this collection of results and an entry point for
those interested in using topological Ramsey space techniques to gain finer
insight into ultrafilters satisfying weak partition relations.
