Network slicing is a key technology to support the concurrent provisioning of heterogeneous Quality of Service (QoS) in the 5th Generation (5G)-beyond and the 6th Generation (6G) networks. However, effective slicing of Radio Access Network (RAN) is very challenging due to the diverse QoS requirements and dynamic conditions in the 6G networks. In this paper, we propose a self-sustained RAN slicing framework, which integrates the self-management of network resources with multiple granularities, the self-optimization of slicing control performance, and self-learning together to achieve an adaptive control strategy under unforeseen network conditions. The proposed RAN slicing framework is hierarchically structured, which decomposes the RAN slicing control into three levels, i.e., network-level slicing, next generation NodeB (gNodeB)-level slicing, and packet scheduling level slicing. At the network level, network resources are assigned to each gNodeB at a large timescale with coarse resource granularity. At the gNodeB-level, each gNodeB adjusts the configuration of each slice in the cell at the large timescale. At the packet scheduling level, each gNodeB allocates radio resource allocation among users in each network slice at a small timescale. Furthermore, we utilize the transfer learning approach to enable the transition from a model-based control to an autonomic and self-learning RAN slicing control. With the proposed RAN slicing framework, the QoS performance of emerging services is expected to be dramatically enhanced.