The area of semiparametric statistics is, in comparison to the areas of fully parametric or nonparametric statistics, relatively unexplored and still in full development. Semiparametric models offer a valid alternative for purely parametric ones, that are known to be sensitive to incorrect model specification, and completely nonparametric models, which often suffer from lack of precision and power. A drawback of semiparametric models so far is, however, that the development of mathematical properties under these models is often a lot harder than under the other two types of models. The present project tries to solve this difficulty partially, by presenting and applying a general method to prove the asymptotic properties of estimators for a wide spectrum of semiparametric models. The objectives of this project are twofold. On one hand we will apply a general theory developed by Chen, Linton and Van Keilegom (2003) for a class of semiparametric Z-estimation problems, to a number of novel research ideas, coming from a broad range of areas in statistics. On the other hand we will show that some estimation problems are not covered by this theory, we consider a more general class of semiparametric estimators (M-estimators called) and develop a general theory for this class of estimators. This theory will open new horizons for a wide variety of problems in semiparametric statistics. The project requires highly complex mathematical skills and cutting edge results from modern empirical process theory.