As the key driving force behind quantum computing, quantum algorithms hold the potential to surmount the limitations of classical computation and achieve exponential speedups. Since the late 20th century, with the theoretical groundwork laid by early algorithms from Shor and Grover, quantum algorithms have experienced rapid advancements in fields such as physical simulation, machine learning, cryptanalysis, and combinatorial optimization. This has led to the development of a comprehensive framework that ranges from theoretical paradigms to practical explorations of algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era. This article offered a systematic review of the evolution of quantum algorithms, examining current major research directions and their technical challenges. These include quantum linear system solvers, quantum many-body and chemical simulations, quantum attacks on symmetric and asymmetric cryptography, post-quantum cryptanalysis, and optimization-focused approaches such as quantum approximate optimization algorithm and quantum annealing. Looking ahead, this article discussed emerging algorithmic frameworks that transcend existing paradigms, the evolution of fault-tolerant and distributed quantum algorithms, and strategic recommendations for national, academic, and industrial development in the field of quantum algorithms. The aim is to provide insights that will advance theoretical innovation and industrial application in quantum computing, ultimately contributing to the establishment of an internationally competitive quantum algorithm system.