This paper provides a global picture about the deployment of networked processing services for genomic data sets. Many current research make an extensive use genomic data, which are massive and rapidly increasing over time. They are typically stored in remote databases, accessible by using Internet. For this reason, a significant issue for effectively handling genomic data through data networks consists of the available network services. A first contribution of this paper consists of identifying the still unexploited features of genomic data that could allow optimizing their networked management. The second and main contribution of this survey consists of a methodological classification of computing and networking alternatives which can be used to offer what we call the Genomic-as-a-Service (GaaS) paradigm. In more detail, we analyze the main genomic processing applications, and classify not only the main computing alternatives to run genomics workflows in either a local machine or a distributed cloud environment, but also the main software technologies available to develop genomic processing services. Since an analysis encompassing only the computing aspects would provide only a partial view of the issues for deploying GaaS system, we present also the main networking technologies that are available to efficiently support a GaaS solution. We first focus on existing service platforms, and analyze them in terms of service features, such as scalability, flexibility, and efficiency. Then, we present a taxonomy for both wide area and datacenter network technologies that may fit the GaaS requirements. It emerges that virtualization, both in computing and networking, is the key for a successful large-scale exploitation of genomic data, by pushing ahead the adoption of the GaaS paradigm. Finally, the paper illustrates a short and long-term vision on future research challenges in the field.