In this dissertation we consider connected topological proper loops L such that their multiplication groups Mult(L) are solvable Lie groups. We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup R by the 2-dimensional nonabelian Lie group or by an elementary fi liform loop. Moreover, if the group Mult(L) has dimension at most 6, then the loop L is centrally nilpotent of class two. We determine the structure of solvable Lie groups which are multiplication groups of 3-dimensional topological loops. We find that there are seven classes of 6-dimensional solvable indecomposable Lie algebras with 5-dimensional nilradical which are the Lie algebras of the multiplication groups Mult(L) of 3-dimensional topological loops L. Among the 6-dimensional solvable indecomposable Lie algebras having 4-dimensional nilradical there are three classes which are Lie algebras of the groups Mult(L). We give the 18 families of decomposable solvable Lie algebras with 1-dimensional centre which are the Lie algebras of Mult(L). Among the 6-dimensional Lie algebras having 2-dimensional centre there are 9 families which can be realized as the Lie algebra of the group Mult(L) of a 3-dimensional connected simply connected topological proper loop L.