Within the realm of approach spaces the authors develop, via the use of suitable nearness concepts, a completion theory and apply this to complete quasimetric spaces. In particular, for approach spaces \(X\) they introduce the concepts of near collections, clusters (= maximal near collections) and completeness (i.e., every cluster converges) and construct a specific completion \(X^*\). Metric spaces and, more generally extended pseudo-metric (for short: generalized quasi-metric) spaces can be considered quite naturally as approach spaces. For metric spaces \(X\) the authors' completion \(X^*\) coincides with the familiar one, in particular is the only metric completion of \(X\). However, approach spaces \(X\) may have completions distinct from \(X^*\). Even metric spaces may have distinct quasi-metric completions. Moreover, if \(X\) is quasi-metric then \(X^*\) may even fail to be generalized quasi-metric. For weakly symmetric approach spaces the natural functor from the category of approach spaces to the category of nearness spaces commutes with completion. For limit-regular quasi-metric spaces completeness means that every Cauchy-sequence converges. In the category of \(T_0\) limit-regular insular generalized quasi-metrie spaces the authors' completion provides an epireflection.