  These are (somewhat informal) lectures notes for the CIME summer school
"Geometric Representation Theory and Gauge Theory" in June 2018. In these notes
we review the results and constructions of a series of our joint papers with
H.Nakajima where a mathematical definition of Coulomb branches of 3d N=4
quantum gauge theories (of cotangent type) is given. We also discuss some
further constructions (such as categories of line operators in the
corresponding topologically twisted theories) and briefly discuss a connection
to local geometric Langlands correspondence.
