  We present quantum (and classical) Bianchi I model, with free massless scalar
field, of the Universe. Our model may be treated as the simplest prototype of
the quantum BKL (Belinskii-Khalatnikov-Lifshitz) scenario. The quantization is
done by making use of the nonstandard Loop Quantum Cosmology (LQC). Since the
method is quite new, we present in details its motivation and the formalism. To
make the nonstandard method easily understandable, we include its application
to the FRW model. In the nonstandard LQC, we first solve the Hamiltonian
constraint of the theory at the classical level and find elementary
observables. Physical compound observables are defined in terms of elementary
ones. We find that classical Big Bang singularity is replaced by quantum Big
Bounce transition due to modification of classical theory by holonomy around a
loop with finite size. The energy density of matter fields at the Big Bounce
depends on a free parameter {\lambda}, which value is expected to be determined
from future cosmological observations. The phase space is divided into two
distinct regions: Kasner-like and Kasner-unlike. We use the elementary
observables to quantize volume and directional volume operators in both cases.
Spectra of these operators are bounded from below and discrete, and depend on
{\lambda}. The discreteness may imply a foamy structure of spacetime at
semiclassical level. At the quantum level an evolution of the model is
generated by the so-called true Hamiltonian. This enables introducing a time
parameter valued in the set of all real numbers.
