  Let ${\mathcal P}$ be a family of probability measures on a measurable space
$(S,{\mathcal A}).$ Given a Banach space $E,$ a functional $f:E\mapsto {\mathbb
R}$ and a mapping $\theta: {\mathcal P}\mapsto E,$ our goal is to estimate
$f(\theta(P))$ based on i.i.d. observations $X_1,\dots, X_n\sim P, P\in
{\mathcal P}.$ In particular, if ${\mathcal P}=\{P_{\theta}: \theta\in
\Theta\}$ is an identifiable statistical model with parameter set
$\Theta\subset E,$ one can consider the mapping $\theta(P)=\theta$ for $P\in
{\mathcal P}, P=P_{\theta},$ resulting in a problem of estimation of
$f(\theta)$ based on i.i.d. observations $X_1,\dots, X_n\sim P_{\theta},
\theta\in \Theta.$ Given a smooth functional $f$ and estimators $\hat
\theta_n(X_1,\dots, X_n), n\geq 1$ of $\theta(P),$ we use these estimators, the
sample split and the Taylor expansion of $f(\theta(P))$ of a proper order to
construct estimators $T_f(X_1,\dots, X_n)$ of $f(\theta(P)).$ For these
estimators and for a functional $f$ of smoothness $s\geq 1,$ we prove upper
bounds on the $L_p$-errors of estimator $T_f(X_1,\dots, X_n)$ under certain
moment assumptions on the base estimators $\hat \theta_n.$ We study the
performance of estimators $T_f(X_1,\dots, X_n)$ in several concrete problems,
showing their minimax optimality and asymptotic efficiency. In particular, this
includes functional estimation in high-dimensional models with many low
dimensional components, functional estimation in high-dimensional exponential
families and estimation of functionals of covariance operators in
infinite-dimensional subgaussian models.
