Abstract: The accurate and efficient computation of three-dimensional (3D) field scattered by large objects of arbitrary shape is a fundamental problem in wave-object interaction studies and of great importance in many practical applications. However, existing numerical methods are computationally intensive and impractical for large objects, especially for large dielectric objects. The conventional ray-based methods neglect wave shape property and remain inefficient. Our vectorial complex ray model (VCRM) incorporates wavefront curvature as an intrinsic property of rays to characterize wave convergence and divergence. The wavefront equation that we derived relates the curvatures of incident, reflected, and refracted wavefronts. VCRM enables, in a ray framework, rigorous computation of amplitude and phase at any point along a ray. The diffraction effects inherently lacked in ray model can be addressed by combining the results of VCRM with physical optics. This model has been validated by comparison of fine 3D scattering patterns of ellipsoidal particles with the multilevel fast multipole algorithm, and that of real liquid jet and pendant drops with experimental results. VCRM can calculate the scattering by objects of size parameter as large as 10⁴. Importantly, a simple application of VCRM with physical optics has clarified several longstanding queries in Airy theory of rainbow that have persisted since the 19th century. This paper provides a comprehensive review of the development of VCRM. It offers a concise and clear introduction to its underlying concepts and algorithms from a practical perspective. Through representative applications, including optical imaging, rainbow research, and 3D scattering of ultra-large objects like ellipsoids, real liquid jet, and pendant droplets, the paper illustrates its usage and computational results, and demonstrates in detail the aforementioned advantages. VCRM is applicable to wave interactions with irregular objects of any shape with smooth surface, offering promising applications in electromagnetic computation, freeform optics, computer graphics, and optical measurements in fluid mechanics.