Quantum computing is rapidly advancing as a powerful tool across scientific fields, addressing computational challenges beyond traditional capabilities. This study explores how quantum computing can accelerate solving NP-hard optimization problems, particularly in logistics and finance. It offers an overview of quantum optimization theories and their practical applications, especially on noisy intermediate-scale quantum (NISQ) devices. We compare classical and quantum optimization, highlight the potentials and limitations of quantum annealing and gate-based computing, and explain the quantum approximation optimization (QAO) algorithm and quantum alternating operator ansatz (QAOA) framework. Despite the promising advancements, a clear quantum advantage has not yet been achieved. The paper emphasizes the ongoing challenges and the essential need for more research to develop effective quantum optimization strategies.