  These notes constitute the basis for the lectures given by the author at
Centre de recherches math\'ematiques (CRM) at Universit\'e de Montreal, as part
of the thematic semester on "Mathematical challenges in many-body physics and
quantum information" (September-December 2018). They are intended for
researchers in mathematics who have a background (and an interest) in
probability theory, but may not be familiar with the area of stochastic
analysis, and in particular with stochastic partial differential equations
(SPDEs). Their goal is to give a brief and concise introduction to the study of
SPDEs using the random field approach, an area which has been expanding rapidly
in the last 30 years, after the publication of John Walsh's lecture notes in
1986. These notes do not survey all the developments in this area, but have the
rather modest goal of introducing the readers to the basic ideas, and
(hopefully) spark their interest to learn more about this subject.
