Eilenberg and Steenrod proved that ordinary homology is characterized by five axioms. Later, Atiyah, Hirzebruch and Whitehead observed that there are other families of functors that satisfy the four “most important” axioms. They defined the so called “generalized homology theories” (or “homology theories”) which are examples of stable phenomena in homotopy theory. The concept of a prespectrum was first introduced by Elon Lages Lima in his PhD thesis to study some kinds of stable phenomena, such as Spanier-Whitehead duality and Stable Postnikov invariants. Later, Adams and Boardman proposed the first homotopy category of prespectrums. This was the starting point of the research field called stable homotopy theory. Nowadays stable homotopy categories are fundamental for studying all kind of stable phenomena in homotopy theory, including generalized homology theories and cohomology theories. The goal of the talk is to present some basic results of algebraic topology and give some elementary stable (and unstable) results of homotopy theory. To reach this goal, we shall introduce the concept of derived functors, homotopy colimits and assume two basic theorems: homotopy excision and long exact sequence of homotopy groups. At the end, we shall prove that every prespectrum represents a homology theory.