  This paper presents a measurement of jet fragmentation functions in 0.49
nb$^{-1}$ of Pb+Pb collisions and 25 pb$^{-1}$ of $pp$ collisions at
$\sqrt{s_{NN}} = 5.02$ TeV collected in 2015 with the ATLAS detector at the
LHC. These measurements provide insight into the jet quenching process in the
quark-gluon plasma created in the aftermath of ultra-relativistic collisions
between two nuclei. The modifications to the jet fragmentation functions are
quantified by dividing the measurements in Pb+Pb collisions by baseline
measurements in $pp$ collisions. This ratio is studied as a function of the
transverse momentum of the jet, the jet rapidity, and the centrality of the
collision. In both collision systems, the jet fragmentation functions are
measured for jets with transverse momentum between 126 GeV and 398 GeV and with
an absolute value of jet rapidity less than 2.1. An enhancement of particles
carrying a small fraction of the jet momentum is observed, which increases with
centrality and with increasing jet transverse momentum. Yields of particles
carrying a very large fraction of the jet momentum are also observed to be
enhanced. Between these two enhancements of the fragmentation functions a
suppression of particles carrying an intermediate fraction of the jet momentum
is observed in Pb+Pb collisions. A small dependence of the modifications on jet
rapidity is observed.
