The author introduces the notion of an ε -fundamental retraction which combines conditions that are used in Borsuk's shape theory, the reviewer's approximate shape theory, and Bogatyi's internal shape theory. This concept is used to define a class of compacta under the name of fundamental approximative absolute neighborhood retracts (FAANRs). It includes fundamental absolute neighborhood retracts (FANRs) and approximative absolute neighborhood retracts in the sense of M. H. Clapp (AANR C s) as proper subclasses. The paper presents several results about the FAANRs which are analogous to the corresponding results about FANRs and AANR C s and it gives several examples which are helpful in understanding the properties of this new class of compacta. For example, it is proved that FAANRs coincide with quasi strongly movable compacta. This statement should be compared with Borsuk's theorem (FANRs coincide with strongly movable compacta) and the reviewer's theorem (AANR C s coincide with approximatively movable compacta). Of course, the author's quasi strong movability is a hybrid of strong movability and approximative movability.