Next-generation network infrastructure should support different services and several levels of quality of service (QoS) and resilience. The main requirements for such multiservice networks are flexibility, effective utilization of network resources, and ability to react to traffic demand changes with time. These requisites lead to the fact that next-generation networks should, to some extend, be "self-adapting," to cope with the need of promptly reacting to traffic demand changes, in an effective way, while taking into account the different requirements in terms of QoS. This paper sketches the reference network scenario and discusses the subject of self-adapting networks, attempting to focus on the key issues that need to be solved to realize future infrastructures. As a relevant example, this paper briefly reports a system developed by the authors' laboratory that implements a traffic engineering solution able to fulfill the mentioned requirements. This system could be regarded as a first concrete step toward the realization of self-adapting networks. In addition, some relevant results that were obtained either by simulation or by experiments on a real test bed are reported to discuss the main characteristics of such a system and assess the feasibility of the concept. Finally, the paper reviews the main hot issues that need to be addressed, in the authors' opinion, by the international research community.