While parametric moment restriction models (MRMs) have been studied extensively, this chapter mainly focuses on semiparametric MRMs (SMRMs) where nonparametrically unknown functions coexist with Euclidean parameters. Two popular estimation methods are semi-nonparametric (SNP) and sieve minimum distance (SMD) methods. To deal with big-data issues, high dimensional SMRMs for cross-sectional and panel data are investigated and estimated by series methods; some new identification conditions are given for factors and factor loadings in panel data models. Several new testing statistics are proposed for over-identification issue. Monte Carlo experiments are conducted to verify the theoretical results.