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This survey gives a comprehensive review of recent advances related to the topic of VoIP QoE (Quality of user' Experience). It starts by providing some insight into the QoE arena and outlines the principal building blocks of a VoIP application. The sources of impairments over data IP networks are identified and distinguished from signal-oriented sources of quality degradation observed over telecom networks. An overview of existing subjective and objective methodologies for the assessment of the QoE of voice conversations is then presented outlining how subjective and objective speech quality methodologies have evolved to consider time-varying QoS transport networks. A description of practical procedures for measuring VoIP QoE and illustrative results is then given. Utilization methodology of several speech quality assessment frameworks is summarized. A survey of emerging single-ended parametric-model speech quality assessment algorithms dedicated to VoIP service is then given. In particular, after presenting a primitive single-ended parametric-model algorithm especially conceived for the evaluation of VoIP conversations, new artificial assessors of VoIP service are detailed. In particular, we describe speech quality assessment algorithms that consider, among others, packet loss burstiness, unequal importance of speech wave, and transient loss of connectivity. The following section concentrates on the integration of VoIP service over mobile data networks. The impact of quality-affecting phenomena, such as handovers and CODEC changeover are enumerated and some primary subjective results are summarized. The survey concludes with a review of open issues relating to automatic assessment of VoIP.