Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=γ(a,x)/Γ(a)$ and $Q(a,x)=Γ(a,x)/Γ(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability. The analytical approach from earlier literature is summarized and new initial estimates are derived for starting the inversion algorithms. The performance of the associated software to our algorithms (the Fortran 90 module {\bf IncgamFI}) is analyzed and compared with earlier published algorithms.