Robust and semiparametric statistics are of the same historical origin and largely employ the same locally asymptotically normal framework. In our talk, we consider he following more intrinsic connections of both fields: 1) Robust influence curves for semiparametric models with infinite dimensional nuisance parameter; for example, for semiparametric regression (Cox), and mixture models (Neyman--Scott). 2) Adaptiveness in the sense of Stein's necessary condition of robust neighborhood models and estimators with respect to a finite dimensional nuisance parameter; for example, location, linear regression, and ARMA. 3) Semiparametric treatment of gross error deviations from an ideal model as an infinite dimensional nuisance parameter, by projection on balls; for testing, an asymptotic version of the Huber--Strassen maximin result is thus obtained. 4) Uniform and nonuniform asymptotic normality of robust and adaptive estimators, respectively, in regression and time series models. 5) Fragility of optimal one-sided tests and confidence limits obtained for convex tangent cones, by projection on cones, as opposed to stability of corresponding procedures, even two-sided, for linear tangent spaces. 6) The unknown neighborhood radius as a nuisance parameter in robustness. The investigation relies on asymptotic techniques and on numerical evaluations of robust estimates. The construction under 1) of robust estimates which are adaptive, an appropriate LAM estimation bound for balls under 3), and the minimization of the norm under 5) on a certain restricted set of differences of tangents from a cone are examples of challenging open problems.