The Capacitated Vehicle Routing Problem (CVRP) is a core NP-hard problem in the field of combinatorial optimization. It aims to plan optimal routes for a fleet of vehicles with uniform capacity, serving a set of customers with specific demands from a single depot, while minimizing the total travel distance. Due to its extensive applications in logistics, distribution, and supply chain management, CVRP has attracted significant research attention. Theoretically, the problem has been proven to be APX-hard, and in general metric spaces, approximate solutions of arbitrary precision cannot be obtained unless P=NP. These inherent complexities highlight the importance of developing approximation algorithms-finding solutions with provable performance guarantees in polynomial time. This paper aims to provide a systematic and comprehensive survey of the research progress on CVRP approximation algorithms. The paper first strictly defines CVRP and its key variants, and elucidates the sources of its fundamental computational complexity. Subsequently, the article deeply analyzes the core algorithmic frameworks and technical schools of thought in this field, including: the Iterated Tour Partitioning (ITP) framework that laid the foundation of the field and its latest improvements; the evolution of approximation schemes (PTAS/QPTAS) for geometric spaces such as Euclidean space; and modern algorithms for structured graphs like trees, planar graphs, and graphs with bounded highway dimension, with a particular focus on techniques based on metric embedding and linear programming relaxation. Finally, this paper summarizes the current best approximation ratios for various problem settings and systematically outlines the core unresolved open problems in the field, pointing out directions for future research.