Point cloud registration (PCR) is a key problem for robotics, autonomous driving, and other applications. Constructing generalizable 3D descriptors and determining whether a 3D descriptor is in the overlapping area are challenging tasks in PCR. Despite the fast evolution of learning-based 3D descriptors, existing methods are either sensitive to rigid transformation and scenario changes, or can not find the appropriate descriptors in the overlapping area, which means their generalization and descriptive ability are not enough for practice applications. To solve these problems, we propose a novel neural network, named G3DOA, which jointly learns generalizable rotation-invariant 3D descriptors and their overlap scores (representing the probability in the overlapping area), to enhance the generalization ability of the descriptors across different data collected by various laser sensors. To ensure the rotation and scale invariance of the point cloud in the input stage, we estimate the Local Reference Frame (LRF) of local patches and normalize the coordinates. To learn generalizable and distinctive descriptors, we propose a novel Cylindrical LRF convolution module with multi-scaled cylindrical shells and neural layers, which hierarchically encodes and aggregates the geometric information in different cylindrical shells. Moreover, to estimate the probability of whether a point is in the overlapping area, we propose an overlap attention module that extracts co-contextual information between the feature encodings of the two point clouds. The experiments show that G3DOA trained only on an indoor dataset can be efficiently generalized to complex outdoor datasets, and the generalization ability of G3DOA outperforms state-of-the-art learning-based 3D descriptors.