Estimation in Semiparametric Models : Some Recent Developments Assume one has to estimate the mean J x P( dx) (or the median of P or any other functional t;;(P)) on the basis ofi.i.d. observations from P. Ifnothing is known about P then the sample mean is certainly the best estimator one can think of. If P is known to be the member of a certain parametric family say {Po: {) E e} one can usually do better by estimating {) first say by {)(n)(.~.) and using J XPo(n)(;r.) (dx) as an estimate for J xPo(dx). There is an intermediate range where we know something about the unknown probability measure P but less than parametric theory takes for granted. Practical problems have always led statisticians to invent estimators for such intermediate models but it usually remained open whether these estimators are nearly optimal or not. There was one exception: The case of adaptivity where a nonparametric estimate exists which is asymptotically optimal for any parametric submodel. The standard (and for a long time only) example of such a fortunate situation was the estimation of the center of symmetry for a distribution of unknown shape. A Survey of basic theory 1. Tangent spaces and gradients 2. Asymptotic bounds for the concentration of estimator sequences 3. Constructing estimator—sequences 4. Estimation in semiparametric models 5. Families of gradients 6. Estimating equations B Semiparametric families admitting a sufficient statistic 7. A special semiparametric model 8. Mixture models 9. Examples of mixture models L Auxiliary results References Notation index.