Stable homotopy theory is that part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor.
A founding result was the Freudenthal suspension theorem, which states that for a given CW-complex $X$ the $(n+i)$-th homotopy group of its $i$-th iterated suspension, $π_{n+i} (Σ^iX)$, becomes stable (i.e., isomorphic after further iteration) for large but finite values of $i$.