Abstract: In this thesis Löwner slit mappings and generalizations are treated analytically and algebraically with the outlook 

solution of Löwner differential equation and extremal points. Analytic characterization of K(t) and Löwner tree is done 

based on Littlewood principle and reflection principle. The extremal properties are treated using variational method and 

Krein-Millman theorem. From Algebraic point of view Löwner semigroup is investigated with its infinitesimal generator 

to give proofs of certain results in this direction making use of Hille-Yosida theorem: A topological aspect is investigated to 

give a characterization of S and L for separability and topological basis. We end up by remarks concerning the role of 

Riesz Representation theorem from measure theoretical point of view.