General Relativity (GR) remains the cornerstone of gravitational physics, providing remarkable success in describing a wide range of astrophysical and cosmological phenomena. However, several challenges underscore the urgent need to explore modified gravity theories. GR struggles to reconcile with quantum mechanics, fails to provide fundamental explanations for dark matter and dark energy, and faces limitations in describing extreme regimes such as black hole singularities and the very early universe. This review provides an organized perspective on modified gravity theories by classifying them based on the principles of GR they preserve or violate. Specifically, we consider three broad categories: (1) metric theories that uphold local Lorentz invariance (LLI) and gauge invariance, (2) theories that break gauge invariance, LLI, or parity, and (3) beyond-metric theories that violate the Einstein's equivalence principle (EEP). This classification highlights the underlying assumptions of GR that these theories challenge or extend, providing a framework for understanding their motivations and implications. The review also discusses the current and upcoming experimental and observational tests of GR, including those probing its foundational principles, such as LLI, gauge invariance, and EEP. For each class of modified theories, we examine their ability to address critical open questions in cosmology and black hole physics. These include their potential to explain the accelerated expansion of the current universe, the nature of dark matter, and deviations in black hole dynamics from GR predictions. This review aims to provide a structured understanding of modified gravity theories and their observational implications in the multimessenger era by focusing on the principles preserved or violated. [abridged]
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