This paper provides a comprehensive review of quantum machine learning (QML), covering both the Noisy Intermediate-Scale Quantum (NISQ) era and fault-tolerant quantum computing (FTQC). It discusses fundamental concepts, algorithms, and statistical learning theory relevant to QML. The review highlights the challenges and opportunities in QML, including the use of variational quantum algorithms (VQA), quantum neural tangent kernels, and quantum landscapes. It also addresses the limitations of NISQ devices, the importance of quantum error correction, and the potential of FTQC for exponential speedups in machine learning. The paper emphasizes the role of classical shadow formalism, quantum random access memory, and shadow tomography in QML. It discusses the trade-offs between classical and quantum approaches, the challenges of barren plateaus, and the potential of quantum algorithms like the Harrow-Hassidim-Lloyd (HHL) algorithm. The review concludes with a discussion of the future of QML, including the potential for fault-tolerant quantum computing to enable more powerful machine learning applications.This paper provides a comprehensive review of quantum machine learning (QML), covering both the Noisy Intermediate-Scale Quantum (NISQ) era and fault-tolerant quantum computing (FTQC). It discusses fundamental concepts, algorithms, and statistical learning theory relevant to QML. The review highlights the challenges and opportunities in QML, including the use of variational quantum algorithms (VQA), quantum neural tangent kernels, and quantum landscapes. It also addresses the limitations of NISQ devices, the importance of quantum error correction, and the potential of FTQC for exponential speedups in machine learning. The paper emphasizes the role of classical shadow formalism, quantum random access memory, and shadow tomography in QML. It discusses the trade-offs between classical and quantum approaches, the challenges of barren plateaus, and the potential of quantum algorithms like the Harrow-Hassidim-Lloyd (HHL) algorithm. The review concludes with a discussion of the future of QML, including the potential for fault-tolerant quantum computing to enable more powerful machine learning applications.