Long-period fiber grating (LPFG) is a kind of wide-range transmission passive photonic device with extensive applications in the field of fiber communication and fiber sensing. In this review, from the angle of refractive index spatial modulation, we extract three characteristic parameters of LPFG:grating period length, index modulated depth and normal orientation of grating plane, and classify LPFG as two types:uniform LPFG (none of these three parameters changes) and nonuniform LPFG (at least one of them changes), and analyze the deficiency of LPFG, including larger size than fiber Bragg grating, no reflection peak, too large bandwidth, polarization loss from single-side exposure, etc. We define the concept of novel LPFG (NLPFG) as the LPFGs based on general LPFG but having new structures and new characters by importing new factors from different aspects, like grating formed mechanism, grating structure, making material, processing technique, application performance, etc. Then we point out that the research significance of NLPFG lies in improving and exploring its real usable property, and making it practical by overcoming the defects of general LPFG in structure, property and application. We expound new techniques of LPFG fabrication, such as multi-exposure, apodized exposure, outfield action, coating and filling, fiber incised and welded, multi-dimensional modulation, and show some NLPFG examples written with these techniques. We build the spatial model of NLPFG to expand the refraction index modulation region from only fiber core to both core and cladding, and to correctly mark the direction of grating plane with tilted angle and azimuth angle. On this basis, we propose the design theory of NLPFG by adding those two angles into the coupling mode coefficient and solving the coupling mode equation. We also expound three different NLPFG design processes, as the direct design to start from given factors of grating, the reserve design to calculate the factors back from expected function or spectrum, and the direct-reserve design combined by them. Meanwhile, we introduce some typical design methods of NLPFG, like geometrical structure changed method, materials changed method, medium coated and embedded method, etc. In addition, we review the recent fabrication and typical application of NLPFG, then introduce different LPFG devices based on excentric core LPFG, multi-core LPFG, few-mode LPFG, stagger LPFG, mismatched LPFG, over-melted LPFG, phase-shift LPFG, tuning LPFG, coupled LPFG and cascaded LPFG, and show their sensing applications in strain, twisting, bending, temperature, displacement, gas concentration and biology. Finally, we provide a developing prospect of the research on NLPFG and give three possible means to improve the research, as innovating new gating structures, exploring new design methods and developing new fabrication techniques.