In the present paper, we introduce Z2-braids and, more generally, G-braids for an arbitrary group G. They form a natural group-theoretic counterpart of G-knots, see [2]. The underlying idea, used in the construction of these objects — decoration of crossings with some additional information — generalizes an important notion of parity introduced by the second author (see [1]) to different combinatorically–geometric theories, such as knot theory, braid theory and others. These objects act as natural enhancements of classical (Artin) braid groups. The notion of dotted braid group is introduced: classical (Artin) braid groups live inside dotted braid groups as those elements having presentation with no dots on the strands. The paper is concluded by a list of unsolved problems.