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Let $X$ be a reduced $n$-dimensional complex space, for which the set of singularities consists of finitely many points. If $X'\subseteq X$ denotes the set of smooth points, the author considers a holomorphic vector bundle $E\to X' \setminus A$, equipped with a Hermitian metric $h$, where $A$ represents a closed analytic subset of complex codimension at least two. The results, surveyed in this paper, provide criteria for holomorphic extension of $E$ across $A$, or across the singular points of $X$ if $A =\varnothing$. The approach taken here is via the metric $h$, and in particular via the $L^2$-theory of the Cauchy-Riemann equation on a punctured neighbourhood for differential $(p,q)$-forms with coefficients in $E$ .
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