Modern interactive and data-intensive applications must operate under demanding time constraints, prompting a shift toward the adoption of specialized software and hardware network acceleration technologies. This specialization, however, poses significant scalability, flexibility, security, and economic sustainability challenges for application developers. Cloud computing holds the potential to overcome these obstacles by offering the cost-effective option to access specialized acceleration technologies through standard cloud interfaces. Nevertheless, that integration is still challenging for cloud providers. In the cloud, physical resources are hidden behind a virtualization layer, whereas acceleration technologies make applications directly interact with the hardware. To bridge this gap, recent literature explores the possibility of empowering cloud platforms with accelerated networking as a commodity, thus offering the innovative option of Network Acceleration as a Service. This survey reviews these recent research efforts by adopting popular technologies like XDP, DPDK, and RDMA as a reference. To organize the surveyed research in a comprehensive framework, we identify four key aspects that pose critical problems to the integration of acceleration options in cloud computing -access interfaces, virtualization techniques, serviceability, and security -and systematically discuss the associated challenges. Then, we present the issues to be further addressed and outline the most promising research directions for the full integration of network acceleration within next-generation cloud computing platforms.