Experimental realization of a universal set of quantum logic gates is the central requirement for the implementation of a quantum computer. In an 'all-geometric' approach to quantum computation(1,2), the quantum gates are implemented using Berry phases(3) and their non-Abelian extensions, holonomies(4), from geometric transformation of quantum states in the Hilbert space(5). Apart from its fundamental interest and rich mathematical structure, the geometric approach has some built-innoise-resilience features(1,2,6,7). On the experimental side, geometric phases and holonomies have been observed in thermal ensembles of liquid molecules using nuclear magnetic resonance(8,9); however, such systems are known to be non-scalable for the purposes of quantum computing(10). There are proposals to implement geometric quantum computation in scalable experimental platforms such as trapped ions(11), superconducting quantum bits(12) and quantum dots(13), and a recent experiment has realized geometric single-bit gates in a superconducting system(14). Here we report the experimental realization of a universal set of geometric quantum gates using the solid-state spins of diamond nitrogen-vacancy centres. These diamond defects provide a scalable experimental platform(15-17) with the potential for room-temperature quantum computing(16-19), which has attracted strong interest in recent years(20). Our experiment shows that all-geometric and potentially robust quantum computation can be realized with solid-state spin quantum bits, making use of recent advances in the coherent control of this system(15-20).