  In this paper we study fields satisfying $N$-point locality and their
properties. We obtain residue formulae for $N$-point local fields in terms of
derivatives of delta functions and Bell polynomials. We introduce the notion of
the space of descendants of $N$-point local fields which includes normal
ordered products and coefficients of operator product expansions. We show that
examples of $N$-point local fields include the vertex operators generating the
boson-fermion correspondences of type B, C and D-A. We apply the normal ordered
products of these vertex operators to the setting of the representation theory
of the double-infinite rank Lie algebras $b_{\infty}, c_{\infty}, d_{\infty}$.
Finally, we show that the field theory generated by $N$-point local fields and
their descendants has a structure of a twisted vertex algebra.
