It is a challenging problem to provide QoS (quality-of-service) guarantee for the next-generation networks, among which QoSR (QoS routing) is one of the key issues. QoSR seeks to find a feasible path for QoS traffic with two objectives: (1) providing the QoS guarantee for QoS traffic, (2) maximizing the utilization of the whole network. However, some multi-constrained QoSR problems, which are NP-complete, can t be resolved accurately in polynomial time. Thus heuristics are studied extensively in recent years. Based on the weighted graph netword model and QoS metrics, the basic concept of QoSR is introduced and the hot issues of the unicast QoSR heuristics are analyzed in this paper. QoSR algorithms can be classified into the several classes: polynomial non-heuristics, pseudo-polynomial non-heuristics, QoS metrics restriction, path subspace search, QoS metrics relation, cost functions study and possibility solution. After typical algorithms are analyzed, the summary and conparison are given for each of the above classes. The validity of algorithms is dissected in detail, including the routing loop problem, stale network state information, the impact of the network topologies and traffic models. Based on the validity analysis, a summary is given for the QOSR algorithms bases on probability models. At last, key issues are pointed out to be further studied in the QoSR field.