The aim of the thesis is to define Steiner system S(t k, v) in simple way and explain the existence condition for STS when k=3, t=2and SQS when k=4, t=3, with some relevant theorems and methods to construct Steiner triple system for every v and Steiner quadruple system for specific value of v. A computer program was developed written in MathCAD 7, running in Pentium II, to the methods and theorems for making the construction easier and faster, In chapter two we produce a new method to find a more than one Steiner triple systems of order 6n+3, n=l,2,3... under non-isomorphic, by using the searching method principle. The second new method we produced is based on extension the triple system to obtain a quadruple system, by adding the appropriate elements to the blocks of the triple system. We end our thesis by listing the programs and the results about the used programs.