  The possibility that primordial black holes (PBHs) form a part of dark matter
has been considered for a long time but poorly constrained in the
$1-100~M_{\odot}$ (or stellar mass range). However, a renewed special interest
of PBHs in this mass window was triggered by the discovery at LIGO of the
merger events of black-hole binaries. Fast radio bursts (FRBs) are bright radio
transients with millisecond duration and high all-sky occurrence rate. Lensing
effect of these bursts has been proposed as one of the cleanest probes for
constraining the presence of PBHs in the stellar mass window. In this paper, we
first investigate constraints on the abundance of PBHs from the latest FRB
observations for both the monochromatic mass distribution and three other
popular extended mass distributions (EMDs). We find that constraints from
currently public FRB observations are relatively weaker than those from
existing gravitational wave detections. Furthermore, we forecast constraining
power of future FRB observations on the abundance of PBHs with different mass
distributions of PBHs and different redshift distributions of FRBs taken into
account. Finally, We find that constraints of parameter space on EMDs from
$\sim10^5$ FRBs with $\overline{\Delta t}\leq1 ~\rm ms$ would be comparable
with what can be constrained from gravitational wave events. It is foreseen
that upcoming complementary multi-messenger observations will yield
considerable constraints on the possibilities of PBHs in this intriguing mass
window.
