Summary: The paper considers a wide class of semiparametric problems with a parametric part for some covariate effects and repeated evaluations of a nonparametric function. Special cases in our approach include marginal models for longitudinal or clustered data, conditional logistic regression for matched case-control studies, multivariate measurement error models, generalized linear mixed models with a semiparametric component, and many others. We propose profile kernel and backfitting estimation methods for these problems, derive their asymptotic distributions and show that in likelihood problems the methods are semiparametric efficient. Although generally not true, it transpires that with our methods profiling and backfitting are asymptotically equivalent. We also consider pseudolikelihood methods where some nuisance parameters are estimated from a different algorithm. The methods proposed are evaluated by using simulation studies and applied to Kenya haemoglobin data.