Abstract: In geometry and topology, as well as in applications of Mathematics to Physics and other areas, one often deals with a system of differential equations and inequalities. By replacing derivatives of unknown functions by independent functions one gets a system of algebraic equations and inequalities. The solvability of this algebraic system is necessary for the solvability of the original system of differential equations. It was a surprising discovery in the 1950-60s that there are geometrically interesting classes of systems for which this condition is also sufficient. This led to counter-intuitive results, like Steven Smale’s famous inside-out "eversion" of the sphere or John Nash’s isometric (i.e. preserving lengths of all curves) embedding of the unit sphere into a ball of an arbitrary small radius. Since that time many more examples of this phenomenon continue to be discovered.