This M.Sc. thesis is based on the following paper • Ghahramani, F., Loy, R.J., Approximate amenability of tensor products of Banach algebras, J. Math. Anal. Appl ??? (????) ???-??? The concept of amenability for a Banach algebra, introducted by Johnson in [??], has proved to be of enormous importance in Banach algebra theory. In [??], and subsequently in [??], several modifications of this notion were introduced, in particular that of approximate amenability; and much work has been done in the last ?? years or so, [??, ?, ?, ??, ?, ??, ??], for example. See also [??] for a recent survey.