Quality of service (QoS) provisioning in next-generation mobile communications systems entails a deep understanding of the delay performance. The delay in wireless networks is strongly affected by the traffic arrival process and the packet service process, which in turn depends on the medium access mechanisms and the signal-to-interference-plus-noise ratio (SINR). In this work, we analyze the delay of large wireless networks, where the sources are distributed as a Poisson point process. We provide analytical expressions for the service process and derive delay violation probability bounds using tools from stochastic network calculus and stochastic geometry. Furthermore, we analyze the delay performance under statistical queueing constraints imposed as limitations on buffer violation probabilities. The impact of QoS requirements, network geometry and spatial density of interferers on the throughput and delay performance is identified. Our results provide useful guidelines for resource allocation and system dimensioning to guarantee stringent delay requirements.