  This work introduces two new notions of dimension, namely the unimodular
Minkowski and Hausdorff dimensions, which are inspired from the classical
analogous notions. These dimensions are defined for unimodular discrete spaces,
introduced in this work, which provide a common generalization to stationary
point processes under their Palm version and unimodular random rooted graphs.
The use of unimodularity in the definitions of dimension is novel. Also, a
toolbox of results is presented for the analysis of these dimensions. In
particular, analogues of Billingsley's lemma and Frostman's lemma are
presented. These last lemmas are instrumental in deriving upper bounds on
dimensions, whereas lower bounds are obtained from specific coverings. The
notions of unimodular Hausdorff size, which is a discrete analogue of the
Hausdorff measure, and unimodular dimension function are also introduced. This
toolbox allows one to connect the unimodular dimensions to other notions such
as volume growth rate, discrete dimension and scaling limits. It is also used
to analyze the dimensions of a set of examples pertaining to point processes,
branching processes, random graphs, random walks, and self-similar discrete
random spaces. Further results of independent interest are also presented, like
a version of the max-flow min-cut theorem for unimodular one-ended trees and a
weak form of pointwise ergodic theorems for all unimodular discrete spaces.
