  In the present article, we consistently develop the main issues of the Bloch
vectors formalism for an arbitrary finite-dimensional quantum system. In the
frame of this formalism, qudit states and their evolution in time, qudit
observables and their expectations, entanglement and nonlocality, etc. are
expressed in terms of the Bloch vectors -- the vectors in the Euclidean space
$\mathbb{R}^{d^{2}-1}$ arising under decompositions of observables and states
in different operator bases. Within this formalism, we specify for all $d\geq2$
the set of Bloch vectors of traceless qudit observables and describe its
properties; also, find for the sets of the Bloch vectors of qudit states, pure
and mixed, the new compact expressions in terms of the operator norms that
explicitly reveal the general properties of these sets and have the unified
form for all $d\geq2$. For the sets of the Bloch vectors of qudit states under
the generalized Gell-Mann representation, these general properties cannot be
analytically extracted from the known equivalent specifications of these sets
via the system of algebraic equations. We derive the general equations
describing the time evolution of the Bloch vector of a qudit state if a qudit
system is isolated and if it is open and find for both cases the main
properties of the Bloch vector evolution in time. For a pure bipartite state of
a dimension $d_{1}\times d_{2}$, we quantify its entanglement in terms of the
Bloch vectors for its reduced states. The introduced general formalism is
important both for the theoretical analysis of quantum system properties and
for quantum applications, in particular, for optimal quantum control, since,
for systems where states are described by vectors in the Euclidean space, the
methods of optimal control, analytical and numerical, are well developed.
