The paper presents an application of linear and affine Gale diagrams and the secondary fan of a point configuration to chemistry. The authors provide a glossary of corresponding terms from discrete geometry and chemistry and define those terms in a way accessible to readers from both communities; this includes all emphasized terms in this review. Represent a chemical compound by a vector in a real vector space of as many dimensions as there are chemical elements, so that for example the coordinates of the compound H\(_2\)O are \(2\) units on the H-axis and one unit on the O-axis. Then a chemical system consisting of \(m\) phases determines an acyclic configuration of \(m\) (not necessarily distinct) vectors that span a subspace of some dimension \(n\); this number is called the number of components of the system. Because the configuration is acyclic, one can rescale the vectors to obtain an affine configuration \(\mathcal A\) of \(m\) points \(a_1,a_2,\dots, a_m\) in an \((n-1)\)-dimensional hyperplane. Possible chemical reactions between phases then correspond to affine dependencies between the points, and minimal reactions to circuits. At a fixed temperature \(T\) and pressure \(P\), each phase \(a_i\) of the system has a certain Gibbs free energy \(g_i(T,P)\). Projecting the lower convex hull of the \(m\) lifted points \((a_i,g_i(T,P))\in\mathbb R^n\) back to \(\mathbb R^{n-1}\) yields a regular (or coherent) polyhedral subdivision of the convex hull of \(\mathcal A\). The vertex sets of the polytopes in this subdivision are precisely the stable assemblages of the participating phases at that temperature and pressure. The secondary fan \(\mathcal F(A)\) of \(\mathcal A\) partitions the space \(\mathbb R^m\) of possible height vectors in such a way that two vectors from the same polyhedral cone of \(\mathcal F(A)\) induce the same subdivision of \(\mathcal A\). The main observation of the paper is that the chemical phase diagram of the chemical system \(\mathcal A\) is given by the two-dimensional slice of the secondary fan \(\mathcal F(A)\) that is the intersection of \(\mathcal F(A)\) with the image of the Gibbs energy map \(g:\mathbb R^2\to\mathbb R^m\), \((T,P)\mapsto(g_1(T,P),\dots, g_m(T,P))\). If this image \(g(\mathbb R^2)\) is ``sufficiently close'' to a \(2\)-dimensional affine plane and moreover intersects \(\mathcal F(A)\) transversally, then determining the possible topological types of chemical phase diagrams of \(\mathcal A\) reduces to finding all possible ways in which the pointed secondary fan \(\mathcal F'(A)\) (obtained from \(\mathcal F(A)\) by factoring out the common lineality space) can decompose a \(2\)-dimensional plane. In this context, the authors provide a geometric proof of Gibbs' phase rule. Moreover, the correspondence between facets of \(\mathcal F'(A)\) and regular triangulations of \(\mathcal A\) and the fact that the set of all regular triangulations of \(\mathcal A\) is connected by bistellar operations (flips) permits the computationally efficient enumeration of chemical phase diagrams. The authors discuss in detail the cases \(m=n+2\) and \(m=n+3\). In the first case, the pointed secondary fan is itself \(2\)-dimensional, so that under the stated linearity and genericity conditions the chemical phase diagram is topologically equivalent to a linear Gale diagram of \(\mathcal A\). Similiarly, for \(m=n+3\) the possible chemical phase diagrams can be read off from an affine Gale diagram of \(\mathcal A\). The paper closes with some remarks on slopes of reaction curves and bounding the number of possible phase diagrams by counting the number of regions in the complement of a hyperplane arrangement.