TITLE: Motivation of stable homotopy theory
QUESTION [34 upvotes]: A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take 
$$\lbrace X,Y \rbrace = \mathrm{colim}[\Sigma^kX,\Sigma^kY]$$
as a hom-set from $X$ to $Y$. Here $\Sigma X = X \wedge S^1$.
What is the motivation of studying such a category (more widely stable homotopy theory)?
For me, the definition of the hom-set looks very fancy, and I can not see why such a category is so important (or useful to topology).

REPLY [7 votes]: Many facets of homotopy theory arose from the work of J.H.C. Whitehead. Once it was realised that the classification up to homotopy type of say finite simplicial complexes  was essentially a countable problem, the question of arose of trying various special cases, as a step to the general problem. The idea was to model such homotopy types by algebraic models. He and Mac Lane proved that connected pointed homotopy $2$-types (which they called $3$-types) were classified by crossed modules. Whitehead also managed to classify all simply connected $4$-dimensional complexes. He also looked to classify $A^n_2$-complexes, i.e. which were $n$-connected and of dimension $\leqslant n+2$, $n \geqslant 4$. 
Further work on these problems has been done by Baues, and others. 
So one kind of answer is that stable homotopy theory is one approximation to homotopy theory, is convenient for many reasons, as well explained by Dylan Wilson, but there are all sorts of other problems and techniques around, particularly those where the fundamental group plays an essential role.