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Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $\left|\right(XB)\setminus X|$, defined on the subsets $X$ with $\left|X\right|\ge k$ and $|G\setminus (XB\left)\right|\ge k$, where $XB$ is the product of $X$ by $B$. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in coming papers to improve...