Abstarct: In recent years, regression modeling has focused on beyond mean regression by describing more general properties of the response distribution. Particularly in different applications, the primary purpose is to assess the effectiveness of covariates on the tail of the response distribution; detecting factors affecting malnutrition in developing countries or reviews of the best quality of agricultural products are two possible examples in such situations. Two class of regression models for achieving this goal are quantile and expectile models. The expectile regression is fitted using quadratic optimization, while the quantile regression requires the optimization of the objective function baxsed on linear programming. So, the expectile regression could be a better alternative for quantile regression. In this thesis, we compare quantiles and expectiles, and we show how to get quantiles from a set of expectiles. We study the quantile and expectile regression models in detail as well as the problem of crossing of quantile and expectile curves. We also re-state a recently proposed method to estimate non-crossing expectile curves baxsed on penalized splines..We explain the introduced methods by using both real and simulated examples.