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, ∆) with K X + ∆ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of ∆ and the volume of K X + ∆.We further show that all n-dimensional projective klt pairs (X, ∆), such that K X + ∆ is big and nef of fixed volume and such that the coefficients of ∆ 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.