This thesis examines the interaction of hydrides of the group 13 metals aluminium and gallium with transition metal centres. Furthermore, a gallium-based system is developed which activates a wide range of E-H bonds, with the product of H₂ activation found to act as a catalyst for the reduction of CO₂ to a methanol derivative. Chapter 3 details the synthesis of a number of alane and gallane adducts of expanded-ring N-heterocyclic carbene (NHC) ligands, which are more strongly σ-donating and sterically shielding analogues of classical NHCs. These NHC adducts are found to be apposite for the formation of σ-alane and σ-gallane complexes at group 6 metal carbonyl fragments, which has allowed the characterisation of the first κ² σ-gallane complexes. The attempted formation of a terminally coordinated κ³ σ-alane complex leads instead to the isolation of a novel dinuclear cluster featuring both μ:κ¹,κ¹ and μ:κ²,κ² coordination to Mo(CO)₃ units. The work presented in Chapter 4 probes the interaction of the β-diketiminate stabilised gallane Dipp₂NacNacGaH₂ with transition metal carbonyls. Far from simply mimicking the chemistry of the alane congener Dipp₂NacNacAlH₂, which forms simple κ¹ and κ² σ-alane complexes, the gallane shows a marked propensity towards dehydrogenation and formation of direct M-Ga(I) bonds. This represents a rare mode of reactivity among group 13 hydrides, being unprecedented beyond boron chemistry, and provides a new route to M-Ga bond formation. Experimental and computational investigations of the mechanism suggest that initial Ga-H oxidative addition is facile, and is generally followed by rate-limiting loss of H₂. The reaction of Dipp2NacNacAlH2 with Co₂(CO)₂ is shown to yield an unusual alane complex which displays an unprecedented degree of Al-H activation in a σ-alane complex. Chapter 5 represents an extension of the work described in Chapter 5, investigating the interaction of Dipp₂NacNacMH₂ (M = Al, Ga) with cationic group 9 transition metal fragments supported by ancillary phosphine ligands. While attempts to isolate unsupported, cationic σ-alane complexes prove unsuccessful, Dipp₂NacNacGaH₂ readily binds to cationic rhodium and iridium centres, forming the first cationic σ-gallane complexes as well as cationic gallylene complexes resulting from complete Ga-H oxidative addition. The extent of Ga-H bond activation is shown to be markedly dependent on the nature of the phosphine co-ligands. In particular, a series of rhodium complexes is reported which represents snapshots of the oxidative addition process, from a Rh(I) σ-gallane complex to a Rh(III) gallylene dihydride, with two further complexes which are on the cusp of these two oxidation states. Described in Chapter 6 are the synthesis and reactivity studies of an ambiphilic system, Dipp₂NacNac′Ga(^tBu), featuring a three-coordinate gallium centre supported by a deprotonated NacNac ligand. The combination of this electrophilic gallium centre with the highly nucleophilic exocyclic alkene functionality facilitates the cooperative activation of protic, hydridic and apolar E-H bonds. Accordingly, molecules including H₂, NH₃, H₂S and SiH4 may be cleaved under mild conditions. Moreover, the hydride product of H₂ activation is shown to be a competent catalyst in conjunction with HBpin for the reduction of CO₂ to the methanol derivative MeOBpin.