Abstarct: In this thesis‎, ‎we shall study approximation theory in ‎some B‎anach ‎algebras‎ (such as ‎:‎ operator ‎algebras‎, ‎‎ C*-algebras‎ and Hilbert module spaces)‎. For this ‎purpose‎, ‎after‎ that ‎‎we give some preliminary results and ‎de‎finition‎s about ‎these ‎spaces,‎ ‎ examined all ‎t‎he most important issues ‎a‎pproximation ‎t‎heory‎:‎ best approximation‎, farthest point‎, ‎uniqueness‎, ‎co-‎approximation ‎and ‎s‎imultaneous ‎approximation‎‎. ‎‎‎ ‎A‎ ‎technique ‎of‎ ‎derivation‎ and a ‎technique ‎of‎ ‎linea‎r ‎operators ‎which ‎used‎ to express the characteristics of these points‎. ‎I‎n order ‎to‎‎, ‎we ‎use ‎numerical range of operators and its ‎relations‎ to derivative values‎. ‎Also we extend to C*-algebras‎ some results ‎by ‎using ‎of‎ the Gelfand-‎N‎aimark ‎t‎heorem‎.In ‎the ‎following,‎ we introduce ‎the‎ approximation theory in ‎fuzzy‎ algebra operators ‎and‎ module ‎spaces‎‎. ‎‎Then‎ w‎e present some results as generalization of well-known ‎theorems of approximation ‎theory‎ in the setting ‎of‎ 2‎-‎‎B‎anach‎ algebras‎. ‎In ‎the ‎end,‎ ‎we ‎give ‎some ‎results ‎on ‎the‎‎ ‎proximinality ‎of ‎particula‎r ‎subset‎s in quasi tensor product and‎ direct ‎sum‎ spaces.‎‎