Coulomb branches of 3-dimensional gauge theories and related structures

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.