This survey traces the evolution of quantum amplitude amplification from its geometric origins in Grover’s algorithm to its algebraic formulation within the Quantum Singular Value Transformation (QSVT) framework. We develop a unified mathematical formalism intended to reduce the historical fragmentation of these algorithms. By establishing a consistent notational foundation early on, framing unstructured search as rotation within an invariant two-dimensional subspace, and following the progression through oblivious, fixed-point, and distributed amplification schemes, we aim to clarify how these methods can be interpreted within the broader language of singular-value polynomial synthesis. We additionally introduce a variational template for amplitude amplification based on parameterized phase rotations, connecting amplification primitives to hybrid quantum-classical algorithms suitable for near-term quantum architectures. To support the pedagogical objectives of this survey, we provide a companion website hosting interactive visualizations and implementation details at QuantumAmplitudeAmplification.