Topology of symplectic manifolds is nowadays a subject of intensive development. The simplest examples of such manifolds are Kähler manifolds and an important property of the latter is their formality. Thus, a possible way of constructing symplectic manifolds with no Kähler structure is to find such ones which are not formal. M. Fernández und V. Muñoz considered in this decade the question of the geography of non-formal compact manifolds. Given a positive integer m and a natural number b, they showed whether or not there are m-dimensional non-formal compact manifolds with first Betti number b. The aim of this thesis is to answer the same question for compact symplectic manifolds. After setting the scene in the first chapter, this is done in the second one - except for the six-dimensional case with b = 1. The third chapter deals with solvmanifolds, especially with those of dimension less than or equal to six, because I hoped to find the missing example among them, and in fact there is a six-dimensional symplectic solvmanifolds which is non-formal and satisfies b = 1. Besides formality, compact Kähler manifolds have even odd-degree Betti numbers and they satisfy the so-called Hard Lefschetz condition. To end Chapter 3, I deal with relations between this three properties for symplectic solvmanifolds. I am able to give an answer to two questions that had remained open in an article of R. Ibáñez, Y. Rudiak, A. Tralle and L. Ugarte. Furthermore, in the last chapter I prove a result that allows an easy computation of the higher homotopy groups of a class of spaces containing all nilpotent ones. Without giving a proof, St. Halperin stated it in the introduction of his book Lectures on Minimal Models