  Modelling diffusion processes in heterogeneous media requires addressing
inherent discontinuities across interfaces, where specific conditions are to be
met. These challenges fall under the purview of Mathematical Analysis as
\emph{transmission problems}. We present a pa\-no\-ra\-ma of the theory of
transmission problems, encompassing the seminal contributions from the 1950s
and subsequent developments. Then we delve into the discussion of regularity
issues, including recent advances matching the minimal regularity requirements
of interfaces and the optimal regularity of the solutions. A discussion on free
transmission problems closes the survey.
