Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory of phase conversion is studied in finite volumes. For the conversion time one finds the relationship $\tau_{\rm con}/\tau_{\rm nu} =1+f_d(q)$. Here $d$ is the space dimension, $\tau_{\rm nu}$ the nucleation time in the volume $V$, and $f_d(q)$ a scaling function. Its dimensionless argument is $q$ reduces the original dependence on three variables (nucleation time, expansion velocity, Volume) to just one variable; $f_d(q)$ is calculated for d = 1, 2 and 3. The often considered limits of phase conversion via either a nucleation or a spinodal process are found to be volume-size dependent concepts, governed by simple power laws for $f_d(q)$. In the infinite volume limit the nucleation region disappears (i.e., the spinodal endpoints collapse to the critical point). Illustrations are given for Potts models.