In this dissertation we have developed the theory of Sagbi bases and Sagbi Groebner bases in a polynomial ring with countably infinite indeterminates over a field and also explored the behavior of Sagbi Groebner bases for ideals in algebras of polynomial ring with finite indeterminate over a field under (de)homogenization technique. We have discussed that the things work in a same manner for Sagbi Groebner bases as in case of Groebner and Sagbi bases