Groups which were almost finitely presented but not finitely presented (FP_2 not F_2) were first constructed by Bestvina and Brady. Since then many more such groups have been constructed by various people using similar techniques. We give the first construction of almost finitely presented subgroups of hyperbolic groups. The key techniques are branched covers and Morse theory. We will discuss some examples of these techniques and describe the how they can be extended to create the subgroups in question.