Mobile networks are facing an unprecedented demand for high-speed connectivity originating from novel mobile applications and services and, in general, from the adoption curve of mobile devices. However, coping with the service requirements imposed by current and future applications and services is very difficult since mobile networks are becoming progressively more heterogeneous and more complex. In this context, a promising approach is the adoption of novel network automation solutions and, in particular, of zero-touch management techniques. In this work, we refer to zero-touch management as a fully autonomous network management solution with human oversight. This survey sits at the crossroad between zero-touch management and mobile and wireless network research, effectively bridging a gap in terms of literature review between the two domains. In this paper, we first provide a taxonomy of network management solutions. We then discuss the relevant state-of-the-art on autonomous mobile networks. The concept of zero-touch management and the associated standardization efforts are then introduced. The survey continues with a review of the most important technological enablers for zero-touch management. The network automation solutions from the RAN to the core network, including end-to-end aspects such as security, are then surveyed. Finally, we close this article with the current challenges and research directions.