  This work covers volume reduction in quantum field theories on a lattice at
large $N$ (number of colors), as first described by Eguchi and Kawai in 1982.
The volume reduction (or volume independence) means that the theory defined on
an arbitrarily small lattice is equivalent in the large-$N$ limit to the theory
on an infinite lattice with the same bare parameters. We analyze the volume
reduction by means of Monte Carlo simulations using the lattice model on a
single site (or a small fixed number of sites) with Wilson fermions in the
adjoint representation, using $N$ up to 60. Most of the results focus on two
flavours of Dirac fermions and the single fermionic flavour is also discussed
where there is a significant difference of behaviour. We find that the
$(Z_N)^4$ center symmetry, necessary for the realization of volume reduction,
is unbroken in the reduced model for a large range of parameters and, in
particular, that the maximum admissible value of the adjoint fermion mass is
non-zero in the large-$N$ limit. We calculate physical quantities, such as the
plaquette, the static quark potential and the eigenvalues of the Dirac
operator. We analyze the finite-$N$ corrections and consider the practicality
of volume-reduced models in supplementing the large-volume calculations.
