Description The parity theory initiated in 2009 by the second named author (V.O. Manturov) argues that if there is a smart way to distinguish between even and odd nodes (crossings, letters) in a way consistent with moves then this allows one to construct functional mappings between objects of the theory, construct various powerful invariants, reduce problems about objects (say knots) to problems about their diagrams, refine many existing invariants. Over the last four years, parity theory has experienced a rapid growth: investigations were undertaken by dozens of scientists worldwide. Various problems in low-dimensional topology were solved by using parity