Gorenstein categories relative to G-admissible triples

We present the notion of Gorenstein categories relative to G-admissible triples. This is a relativization of the concept of Gorenstein category (an abelian category with enough projective and injective objects, in which the suprema of the sets $\{ {\rm pd}(I) \ \text{:} \ I \text{ is injective} \}$ and $\{ {\rm id}(P) \ \text{:} \ P \text{ is projective} \}$ are finite). Such categories turn out to be a suitable setting on which it is possible to obtain hereditary abelian model structures where the (co)fibrant objects are Gorenstein injective (resp., Gorenstein projective) objects relative to GI-admissible (resp., GP-admissible) pairs. Applications and examples of these structures are given. Moreover, we link relative Gorenstein categories with tilting theory and obtain relations between different relative homological dimensions.