  We show that the number of marked minimal models of an n-dimensional smooth
complex projective variety of general type can be bounded in terms of its
volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional
projective klt pair (X,D) with $K_X+D$ big, we show more generally that the
number of its weak log canonical models can be bounded in terms of the
coefficients of D and the volume of $K_X+D$. We further show that all
n-dimensional projective klt pairs (X,D), such that $K_X+D$ is big and nef of
fixed volume and such that the coefficients of D are contained in a given DCC
set, form a bounded family. It follows that in any dimension, minimal models of
general type and bounded volume form a bounded family.
