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Future networks, such as ATM, will support a wide range of multimedia services, such as voice and video, that exhibit diverse traffic characteristics but also require specific quality of service (QoS) guarantees. Most existing approaches that support QoS guarantees (1) assume the existence of a path between the communicating entities and (2) ask the nodes of the path to reserve enough resources to satisfy the requested QoS. The main drawback of these approaches is that they do not address the routing problem, which is closely related to QoS guarantees; if the routing is performed without QoS constraints, several retries may be necessary before the connection is established. Several routing procedures (with QoS constraints) are proposed in the literature; however, they consider only a subset of QoS requirements, e.g. bandwidth or delay, while routing the connection. In this paper, we present a routing procedure which considers different QoS classes; depending on the QoS class the connection belongs to, the procedure selects the "best" path that minimizes the parameters (of the class) while satisfying the whole requirements of the connection. For example, the path with the minimum end-to-end delay which satisfies the delay, the bandwidth and the loss rate requirements is selected in case the connection belongs to delay-sensitive class. The proposed routing procedure is essentially independent of network technologies, such as ATM or stop-and-go networks; however the network nodes should be capable of reserving resources when requested.