  This paper presents recent work on connections between symmetries and
conservation laws. After reviewing Noether's theorem and its limitations, we
present the Direct Construction Method to show how to find directly the
conservation laws for any given system of differential equations. This method
yields the multipliers for conservation laws as well as an integral formula for
corresponding conserved densities. The action of a symmetry (discrete or
continuous) on a conservation law yields conservation laws. Conservation laws
yield non-locally related systems that, in turn, can yield nonlocal symmetries
and in addition be useful for the application of other mathematical methods.
  From its admitted symmetries or multipliers for conservation laws, one can
determine whether or not a given system of differential equations can be
linearized by an invertible transformation.
