This PhD-Thesis which has been worked out within the framework of the FWF Fundamental Research Area S83 Algorithmic number theory and applications deals with the determination of integer solutions of an essential class of algebraic equations. Well known classical problems of this kind are Fermats Last Theorem and the computation of Pythagorean Triples. Nowadays such results have an important field of application within the encryption of data using elliptic curves. In the present work so called Thue equations are examined, i.e., equations of the form F(x,y) = m where F(x,y) is an irreducible homogenous polynomial in x and y of degree at least 3 and m is an integer. In particular the work deals with parameterized families of cubic Thue equations over imaginary quadratic number fields, i.e., particular number fields within the field of complex numbers. To find all solutions x, y in the corresponding ring of integers, the idea is used that solving the above Thue equation is equivalent to determining all elements with small relative norm in a certain number field of degree 6. For that reason as well as for the succeeding treatment of the solutions of the Thue equation, various algebraic auxiliary results had to be developed and extensive computer-aided analysis had to be executed using computer algebra.