Divisible designs graphs (DDGs for short) were introduced in 2011 by Haemers, Kharaghani and Meulenberg as a bridge between graph theory and the theory of (group divisible) designs. Since then, around 20 constructions producing infinitely many DDGs haven been introduced. These constructions make use of many combinatorial and algebraic objects: finite geometries, Hadamard matrices, weighing matrices, designs, Cayley graphs, block matrices, strongly regular graphs and so on. Also, a number of characterisations of divisible design graphs is known. In this talk we will discuss three infinite families of DDGs recently discovered in connection with symplectic spaces. The talk is based on joint works with Anwita Bhowmik, Bart De Bruyn, Willem Haemers and Leonid Shalaginov.