This review provides a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. It begins by introducing quantum noise spectra and methods for their detection, followed by a description of weak continuous measurements and their application to qubit and oscillator state detection. The review then delves into general linear response theory, discussing quantum constraints on noise and the quantum limit on QND detection of a qubit. The core of the review focuses on the quantum limit on linear amplifiers and position detectors, including the standard Haus-Caves derivation of the quantum limit for a bosonic amplifier and its application to electrical circuits. The text also explores various methods for evading the quantum limit through back-action evasion techniques. The review concludes with a discussion of practical considerations for achieving quantum-limited measurements and a summary of key results.This review provides a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. It begins by introducing quantum noise spectra and methods for their detection, followed by a description of weak continuous measurements and their application to qubit and oscillator state detection. The review then delves into general linear response theory, discussing quantum constraints on noise and the quantum limit on QND detection of a qubit. The core of the review focuses on the quantum limit on linear amplifiers and position detectors, including the standard Haus-Caves derivation of the quantum limit for a bosonic amplifier and its application to electrical circuits. The text also explores various methods for evading the quantum limit through back-action evasion techniques. The review concludes with a discussion of practical considerations for achieving quantum-limited measurements and a summary of key results.