In the opinion of the celebrated mathematician and philosopher of science René Thom, chemical concepts such as those of bonds and valence lack scientific content [1]. He pointed out that the vocabulary used by chemists to describe matter at a microscopic level was often ambiguous. In fact most concepts have their origin in models developed at the beginning of the twentieth century and earlier which are, therefore, not fully consistent with quantum mechanics. To reconcile the chemical description of matter with the postulates of quantum mechanics it is necessary to build a mathematical model. This mathematical model is not unique, however, because different spaces (geometrical direct space, momentum space, Hilbert space) and different mathematical theories external to quantum mechanics can be used for this purpose.The choice of the geometrical space has been pioneered by Raymond Daudel with the loge theory [2-4] the applicability of which remained limited to very small systems because the underlying mathematical theory, Shannon's information theory [5], requires evaluation of the N-particle distribution function. The theory of dynamical systems [6, 7] is a powerful method of analysis of space which has been convincingly introduced in chemistry by Richard Bader with the quantum theory of atoms in molecules (QTAIM) [8]. This theory is based on analysis of the gradient field of the electron-density distribution which enables the partitioning of the molecular space into basins associated to each atom.The purpose of this chapter is to provide a digest of the topological theory of chemical bonding with particular emphasis on epistemological aspects. QTAIM has made important contributions to conceptual chemistry, for example the definition of the atom within a molecule, of the bond critical point, of the bond path, and of the charge concentrations or depletions which emerge from analysis of the Laplacian of the charge density LðrÞ and which make a link with the valence shell electron pair repulsion (VSEPR) model [9][10][11][12][13][14][15]. To provide evidence for basins which correspond to bonds and lone pairs, however, one must perform dynamic