  We review the Lagrangian formulation of Noether symmetries (as well as
"generalized Noether symmetries") in the framework of Calculus of Variations in
Jet Bundles, with a special attention to so-called "Natural Theories" and
"Gauge-Natural Theories", that include all relevant Field Theories and physical
applications (from Mechanics to General Relativity, to Gauge Theories,
Supersymmetric Theories, Spinors and so on). It is discussed how the use of
Poincare'-Cartan forms and decompositions of natural (or gauge-natural)
variational operators give rise to notions such as "generators of Noether
symmetries", energy and reduced energy flow, Bianchi identities, weak and
strong conservation laws, covariant conservation laws, Hamiltonian-like
conservation laws (such as, e.g., so-called ADM laws in General Relativity)
with emphasis on the physical interpretation of the quantities calculated in
specific cases (energy, angular momentum, entropy, etc.). A few substantially
new and very recent applications/examples are presented to better show the
power of the methods introduced: one in Classical Mechanics (definition of
strong conservation laws in a frame-independent setting and a discussion on the
way in which conserved quantities depend on the choice of an observer); one in
Classical Field Theories (energy and entropy in General Relativity, in its
standard formulation, in its spin-frame formulation, in its first order
formulation "`a la Palatini" and in its extensions to Non-Linear Gravity
Theories); one in Quantum Field Theories (applications to conservation laws in
Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).
