An experiment has been conducted to study the effects of the inclusion of plant oil in rabbit diets. This study was aimed at evaluating the beneficial effects of the inclusion of maize oil (MO), rich in unsaturated fatty acids (UFAs), compared to palm oil (PO) containing saturated fatty acids (SFAs), on the meat fatty acid (FA) profile. As UFAs are susceptible to rancidity, Curcuma longa (CL), which is known for its antioxidant properties, was also added (3 g/kg) to the diet with two plant oils. CL contains curcuminoids, volatile oils, sugars, proteins, resins and polyunsaturated fatty acids (PUFAs). We also evaluated the influence of CL inclusion in the diet on the FA profile of the meat. Furthermore, the possibility of using these oil-enriched diets and the ability to assimilate CL in rabbits was evaluated by analysing the performance, carcass characteristics and meat quality. At the end of the experiment, there were no significant differences between the groups concerning the live weight, live weight gain, feed consumption, feed efficiency, carcass yield or the percentages of edible organs. The hind legs, forelegs, loins and abdominal wall, breast and ribs, skin and limbs and head were not affected by the oil type or by the inclusion of CL. The chemical composition, pH and oxidative status of the Longissimus dorsi muscle of the rabbits fed the experimental diets were not affected by the oil source or by the CL supplementation. Conversely, it has been shown that it is possible to modify the FA profile of rabbit meat and fat by dietary means. The SFA/PUFA ratio significantly decreased from −18% to −16% in the meat and from −25% to −23% in the perirenal fat of the rabbits fed diets containing MO without or with CL supplementation, respectively, compared to same tissues of the rabbits fed diets containing PO without or with CL supplementation, respectively. Similar trends were found for the atherogenic index, which decreased from −20% to −17% in the meat and from −26% to −23% in the perirenal fat, respectively, and the thrombogenic index, which decreased from −19% to −24% in the meat and from −24% to −23% in the perirenal fat, respectively. CL increased the α-Linolenic acid and PUFA n-3 contents and reduced the vaccenic acid content and the n-6/n-3 ratio in the meat of the rabbits fed the PO or MO diets.