Similar to macroscopic objects, photons can also carry angular momentum, such as spin angular momentum (SAM) and orbital angular momentum (OAM). The two angular momenta contribute to a new structured beam, the vectorial vortex beam (VVB). A VVB has anisotropic wavefront and polarization distributions, thus providing multiple degrees of freedom and showing great potential in lots of advanced domains including quantum technology, laser communications, laser detection, laser processing, high-resolution imaging, and optical tweezers, attracting much attention around the world. The key to employing VVBs in these above scenarios is their generation and recognition with high performance. We review recent advances in generating and diagnosing VVBs in brief. In addition, we also systematically review research on this topic from our team in the past decade and focus more on our representative achievements.

This review consists of three main sections. The basic principles of VVBs are introduced in the first section. The introduction begins with the decomposition of VVBs, as a VVB can be regarded as the coaxial superposition of two scalar beams with opposite SAM and various OAMs. Then, typical representations of VVBs, the hybrid order Stokes parameters, and Poincare spheres are reviewed. Finally, our recent demonstration for VVB mode representation, the four-parameter notation, and its great performance is introduced.

The second section presents recent advances in VVB generation, including extra- and intra- cavity generations. The extra-cavity generation scheme is to transform other beams containing Gaussian beams and Hermit-Gauss beams into VVBs outside a laser resonator, whose principle is based on the decomposition of VVBs. In addition, such a scheme is flexible to produce more complex vectorial vortex fields, including the VVB array and perfect vortex array. By employing programmable devices of the liquid-crystal spatial light modulator, VVBs can also be generated digitally. The intra-cavity generation scheme is to output VVBs from a laser resonator directly. One can place multiple devices or optical elements inside the cavity, thereby leading to the oscillation of high-order transverse modes including VVBs. One of the most common intra-cavity elements is Q-plate, with photon SAM-OAM conversion elements fabricated based on photon spin Hall effect. Such elements can transform the oscillated fundamental mode to vectorial vortex mode and meet with the mode self-consistence in a laser cavity. Furthermore, the spatial light modulator can also be employed as part of the resonator to replace the end mirror to form a "digital laser" and enable VVBs output. Our recent work is also presented emphatically in this section, including eye-safe solid VVB lasers, single frequency VVB lasers, and nonplanar ring oscillator VVB lasers.

The third section presents recent advances in the vectorial vortex mode recognition. Vectorial vortex mode origins from the classical entanglement of SAM and OAM. Thus a VVB is also a total angular momentum (TAM) mode, and vectorial vortex mode recognition is equivalent to TAM measurement. As photon SAM has only two eigenvalues, the key to vectorial vortex mode recognition is to measure OAM distribution. This section introduces more about OAM analysis developed by our team, including universal OAM spectrum analyzer, deep learning-assisted OAM spectrum measurement, and photon OAM sorter. The universal OAM spectrum analyzer is based on the helical harmonic decomposition of beams, which is the definition of the OAM spectrum. Therefore, such an analyzer is universal and appropriate for beams with any patterns. The deep leaning-assisted OAM spectrum measurement is to extract OAM features firstly and analyze the extracted pattern through our developed convolutional neural network, the adjusted EfficientNet. The OAM sorter is accomplished through our developed multi-ring azimuthal-quadratic phase, and supports up to 73 OAM modes.

We hope this review will provide more useful information for people who study VVBs and their applications, and inspire more novel and wonderful ideas.