  In this paper, important concepts from finite group theory are translated to
localities, in particular to linking localities. Here localities are group-like
structures associated to fusion systems which were introduced by Chermak.
Linking localities (by Chermak also called proper localities) are special kinds
of localities which correspond to linking systems; thus they contain the
algebraic information that is needed to study $p$-completed classifying spaces
of fusion systems as generalizations of $p$-completed classifying spaces of
finite groups.
  Because of the group-like nature of localities, there is a natural notion of
partial normal subgroups. Given a locality $\mathcal{L}$ and a partial normal
subgroup $\mathcal{N}$ of $\mathcal{L}$, we show that there is a largest
partial normal subgroup $\mathcal{N}^\perp$ of $\mathcal{L}$ which, in a
certain sense, commutes elementwise with $\mathcal{N}$ and thus morally plays
the role of a ``centralizer'' of $\mathcal{N}$ in $\mathcal{L}$. This leads to
a nice notion of the generalized Fitting subgroup $F^*(\mathcal{L})$ of a
linking locality $\mathcal{L}$. Building on these results we define and study
special kinds of linking localities called regular localities. It turns out
that there is a theory of components of regular localities akin to the theory
of components of finite groups. The main concepts we introduce and work with in
the present paper (in particular $\mathcal{N}^\perp$ in the special case of
linking localities, $F^*(\mathcal{L})$, regular localities and components of
regular localities) were already introduced and studied in a preprint by
Chermak. However, we give a different and self-contained approach to the
subject where we reprove Chermak's theorems and also show several new results.
