I will give a gentle introduction to total positivity and the
theory of Polya frequency (PF) functions. This includes their
spectral properties, basic examples including via convolution,
and a few proofs to show how the main ingredients fit together.
Many classical results (and one Hypothesis) from before 1955
feature in this journey. I will end by describing how PF functions
connect to the Laguerre–Polya class and hence Polya–Schur
multipliers, and mention 21st century incarnations of the latter.