Invariants and pictures : low dimensional topology and combinatorial group theory This book contains an in depth overview of the current state of the recently emerged and rapidly growing theory of Gk n groups picture valued invariants and braids for arbitrary manifolds. Equivalence relations arising in low dimensional topology and combinatorial group theory inevitably lead to the study of invariants and good invariants should be strong and apparent. An interesting case of such invariants is picture valued invariants whose values are not algebraic objects but geometrical constructions like graphs or polyhedra. In 2015 V. O. Manturov defined a two parametric family of groups Gk n and formulated the following principle: if dynamical systems describing a motion of n particles possess a nice codimension 1 property governed by exactly k particles then these dynamical systems possess topological invariants valued in Gk n. The book is devoted to various realisations and generalisations of this principle in the broad sense. The groups Gk n have many epimorphisms onto free products of cyclic groups; hence invariants constructed from them are powerful enough and easy to compare. However this construction does not work when we try to deal with points on a 2 surface since there may be infinitely many geodesics passing through two points. That leads to the notion of another family of groups \\Gamma_n^k which give rise to braids on arbitrary manifolds yielding invariants of arbitrary manifolds Groups. Small cancellations. Greendlinger theorem Braid theory Curves on surfaces. Knots and virtual knots Two dimensional knots and links Parity in knot theories. The parity bracket Cobordisms General theory of invariants of dynamical systems Groups Gk n and their homomorphisms Generalisations of the groups Gk n Representations of the groups Gk n Realisation of spaces with Gk n action Word and conjugacy problems in Gk k+1 groups The groups Gk n and invariants of manifolds The two dimensional case The three dimensional case Open problems.