  The paper discusses inference techniques for semiparametric models based on
suitable versions of inference functions. The text contains two parts. In the
first part, we review the optimality theory for non-parametric models based on
the notions of path differentiability and statistical functional
differentiability. Those notions are adapted to the context of semiparametric
models by applying the inference theory of statistical functionals to the
functional that associates the value of the interest parameter to the
corresponding probability measure. The second part of the paper discusses the
theory of inference functions for semiparametric models. We define a class of
regular inference functions, and provide two equivalent characterisations of
those inference functions: One adapted from the classic theory of inference
functions for parametric models, and one motivated by differential geometric
considerations concerning the statistical model. Those characterisations yield
an optimality theory for estimation under semiparametric models. We present a
necessary and sufficient condition for the coincidence of the bound for the
concentration of estimators based on inference functions and the semiparametric
Cram\`er-Rao bound. Projecting the score function for the parameter of interest
on specially designed spaces of functions, we obtain optimal inference
functions. Considering estimation when a sufficient statistic is present, we
provide an alternative justification for the conditioning principle in a
context of semiparametric models. The article closes with a characterisation of
when the semiparametric Cram\`er-Rao bound is attained by estimators derived
from regular inference functions.
