Quantum Computing is revolutionary paradigm that harnesses the principles of quantum mechanics such as superposition and entanglement to process the information in fundamentally new ways. It has shown potential across various fields, including Cryptography, Materials Science, Machine Learning, Optimization and many others. Among its many applications, Combinatorial Optimization problems stand out as a critical area where Quantum Computing can address problems that are often intractable for classical methods. This paper surveys current quantum approaches including Quantum Annealing, Variational Algorithms, Quantum Approximation Optimization Algorithm (QAOA) and asses their effectiveness in solving Classical Intractable Combinatorial Optimization (CICO) problems such as Max Cut Problem, Traveling Salesman Problem, Graph Coloring and Knapsack Problems. The paper explores the QAOA, approach to solve the CICO problems. Practical applications of quantum optimization approaches are highlighted in the fields such as finance, material science, communication networks. Additionally, the case studies demonstrate the algorithm's efficacy in real world scenarios. However, the paper also discusses the challenges that remain including issues related to noise, scalability and complexity of the algorithms. Finally, the survey discusses the future directions for research, emphasizing the potential of hybrid-quantum classical frameworks and the development of new quantum algorithms, aiming to enhance the practical applications in complex optimization scenarios. This survey provides valuable insights for researchers and practitioners interested in leveraging quantum advantages to solve combinatorial optimization problems effectively.