Let *G* be a finite permutation group. T
he fixed point ratio of an element *x* in <
em>G\, denoted fpr(*x*)\, is the propo
rtion of points fixed by *x*. Fixed point r
atios for finite primitive groups have been studie
d for many decades\, finding a wide range of appli
cations. In this talk\, I will discuss some recent
joint work with Bob Guralnick where we determine
the triples (*G*\,*x*\,*r*) s
uch that *G* is primitive\, *x* has
prime order *r* and fpr(*x*)>\;1/(
*r*+1). This turns out to have some interes
ting applications and we can use it to obtain new
results on the minimal degree and minimal index of
primitive groups. Another application arises in j
oint work with Moreto and Navarro on the commuting
probability of *p*-elements in finite grou
ps.