This thesis is a generalization of the works done by Christensen and Laugesen( [5]). We introduce approximately dual frame in Hilbert spaces. Frames were introduced by Duffin and Schaeffer in their fundamental paper. They used frames as a tool in the study of nonharmonic Fourier series. As the wavelet era began, Daubechies, Grossmann, and Meyer observed that frames can be used to find series expansions of functions in L 2 (R) that are very similar to the expansions using orthonormal bases. This was the time when many mathematicians started to see the potential of the topic. A frame is a sequence of elements {f k } in H , which allows every f in H to be written as an infinite linear combination of elements f k in the frame. However, the corresponding coefficients are not necessarily unique. If {f k } is a frame, then one choice of dual is the canonical dual frame. Let {f k } be a frame with frame operator S, then the frame {S -1 f k } is called the canonical dual frame of {f k }. T he most important frame result, shows that if {f k } is a frame for H , then every element in H has a representation as an infinite linear combination of the frame elements. We aim at a characterization of all dual frames {g k } associated with a given frame {f k }. Unfortunately, it might be cumbersome or even impossible to calculate a dual frame explicitly. One finds only a few infinite dimensional non-tight frames for which a dual has been constructed. This paucity of constructions leads us to seek frames that are closed to dual. Approximately duals are easier to construct than We use perturbation ideas to construct approximately dual frames. There are situations where it is hard to find a dual frame for a given frame {h k } , but where {f k } lies close to a frame {h k } for which a dual frame {g k } is known explicitly. We seek to connect this fact with perturbation theory by asking the following question: if we can find a frame {f k } that is close to {h k } and for which it is possible to find a dual frame {g k } , does it follow that {f k } is an approximate dual of {g k } ? We peresent conditions under which such a frame {g k } is approximately dual to {f k } , both in the case where {g k } is an arbitrary dual of {h k } and for the particular case where it is the canonical dual. Such general Hilbert space estimates might of course be improved for concrete An alternative bound is derived for the rich To illustrate these results, we construct explicit approximately duals of Gabor frames generated by the Gaussian; these approximate duals yield almost perfect reconstruction.