We have proposed a new LCDE method in circular-circular regression, where estimates are obtained from minimizing the sum of perpendicular (shortest) circular distances. This new method is similarly applied to the LCDEs in the inverse circular regressions. A new theorem for the multi-variate CLT of the least circular distance estimators (LCDE) can be found Chapter 4. A new performance comparison criteria, called the relative circular prediction bias (RCPB), is introduced in Chapter 3, with its applications being found in various chapters. As the title of this work explains, our main research work is contained in Chapter 5, where we introduce the asymptotic generalized vonMises distribution, and use in the circular regression models in order to account for a skewness in the circular response variable. Chapter 6 and 7 consist of related and supplementary material. The appendices appearing at the end of our paper include proofs of some propositions, as well as all the data sets used in our work. The chapters were arranged in order to give our readers knowledge and review that might be needed in the following chapters.