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Abstract. Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of'higher KdV equations' are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomologicai nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that 'symmetry' and 'conservation law' are, in some sense, the 'dual' conceptions which coincides in the 'self-dual' case, namely, for Euler-Lagrange equations. Training examples are also given.

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