A nonlinear Wightman field is taken to be a nonlinear map from a linear space of test functions to a linear space of Hilbert space operators, with inessential modifications to other axioms only to the extent dictated by the introduction of nonlinearity. Two approaches to nonlinear quantum fields are constructed and discussed, the first of which, starting from Lagrangian QFT, offers a fresh perspective on renormalization, while the second, starting from linear Wightman fields, provides an extensive range of well-defined nonlinear theories. [For LQP purposes, Section IV is the principal interest.]