This review article provides a comprehensive overview of the physics of Anderson transitions in disordered systems, including both metal-insulator transitions and quantum-Hall-type transitions. The authors emphasize recent developments, such as multifractality of critical wave functions, criticality in power-law random banded matrix models, symmetry classification of disordered electronic systems, and mechanisms of criticality in quasi-one-dimensional and two-dimensional systems. The review covers analytical approaches and advanced numerical simulations, highlighting the progress in understanding the Anderson localization phenomenon and associated quantum phase transitions in disordered electronic systems. Key topics include symmetry classification and universality classes, multifractality of wave functions, quantitative understanding of critical behavior, and criticality in various models and systems. The article also discusses the role of ensemble averaging in determining multifractal spectra and the dimensionality dependence of wave function statistics at the Anderson transition.This review article provides a comprehensive overview of the physics of Anderson transitions in disordered systems, including both metal-insulator transitions and quantum-Hall-type transitions. The authors emphasize recent developments, such as multifractality of critical wave functions, criticality in power-law random banded matrix models, symmetry classification of disordered electronic systems, and mechanisms of criticality in quasi-one-dimensional and two-dimensional systems. The review covers analytical approaches and advanced numerical simulations, highlighting the progress in understanding the Anderson localization phenomenon and associated quantum phase transitions in disordered electronic systems. Key topics include symmetry classification and universality classes, multifractality of wave functions, quantitative understanding of critical behavior, and criticality in various models and systems. The article also discusses the role of ensemble averaging in determining multifractal spectra and the dimensionality dependence of wave function statistics at the Anderson transition.