In this thesis, we present an expanded account of the work done by Yong Zhang. On approximate complementatio in Banach algebras, at the first we define approximately complemented spaces. A suace M of a Banach space X is called approximately complemented i X if there is a net {P ? } of continuous operators from X into M such that P ? ( x ) converges to X uniformly on every compact subset of M . If in addition {P ? } can be chosen to be a bounded net, i.e., there is a constant m such that ? P ? ? ? m for all ? , the M is called bounded approximately complemented in X . Then we show that a suace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal I has a right (resp. left) approximate identity (P ? ) such that, for every compact subset K of I , the net (a.P ? ) (resp. (P ? . a) ) converges to a uniformly for a in K if and only if I is approximately complemented in the algebra. Let A be a Banach algebra, then a left (right) approximate identity (e ? ) ?? D in A is said to satisfy condition (U) if e ?. a (resp. a.e ? ) converges to a uniformly for a in compact sets of A . Let X be a Banach space, B(X) be the operator algebra of continuous operators o X and A be a subalgebra of B(X) . It is shown that A has a left approximate identity satisfying condition (U) if and only if the identity operator on the space X is approximable uniformly on compact sets of X by operators i A.