What are symmetries of nonlinear PDEs and what are they themselves?

The general theory of (nonlinear) partial differential equations originated by S. Lie had a significant development in the past 30-40 years. Now this theory has solid foundations, a proper language, proper techniques and problems, and a wide area of applications to physics, mechanics, to say nothing about traditional mathematics. However, the results of this development are not yet sufficiently known to a wide public. An informal introduction in a historical perspective to this subject presented in this paper aims to give to the reader an idea about this new area of mathematics and, possibly, to attract new researchers to this, in our opinion, very promising area of modern mathematics.