This review provides an overview of the emerging field of moiré materials, which are highly tunable, strongly correlated two-dimensional (2D) materials engineered by the rotational or lattice misalignment of atomically thin crystals. The confluence of topological and electronic correlation effects in moiré materials has led to the discovery of a rich variety of exotic phases, including correlated insulators, unconventional superconductors, and topological phases. The review highlights the contributions of local spectroscopic, thermodynamic, and electromagnetic probes to the understanding of these materials. These techniques have identified many underlying mechanisms of correlated insulators, generalized Wigner crystals, unconventional superconductors, moiré ferroelectrics, and topological orbital ferromagnets. The review also discusses the advantages of local probe techniques, such as scanning tunneling microscopy (STM), angle-resolved photoemission spectroscopy (ARPES), scanning single-electron transistor (SET), and scanning nano-scale superconducting quantum interference device (nano-SQUID) measurements. These techniques provide spatially resolved insights into the electronic structure and phase transitions in moiré materials, revealing complex quantum phases that are difficult to access using global probes. The review covers the discovery of flat electronic bands, cascades of electronic transitions, and the identification of symmetry-broken orders in moiré graphene and transition metal dichalcogenides (TMDs). It also discusses the imaging of correlated insulators, electron crystal phases, and charge-ordered phases in TMDs, emphasizing the importance of local probe techniques in understanding the intricate electronic mechanisms in these materials.This review provides an overview of the emerging field of moiré materials, which are highly tunable, strongly correlated two-dimensional (2D) materials engineered by the rotational or lattice misalignment of atomically thin crystals. The confluence of topological and electronic correlation effects in moiré materials has led to the discovery of a rich variety of exotic phases, including correlated insulators, unconventional superconductors, and topological phases. The review highlights the contributions of local spectroscopic, thermodynamic, and electromagnetic probes to the understanding of these materials. These techniques have identified many underlying mechanisms of correlated insulators, generalized Wigner crystals, unconventional superconductors, moiré ferroelectrics, and topological orbital ferromagnets. The review also discusses the advantages of local probe techniques, such as scanning tunneling microscopy (STM), angle-resolved photoemission spectroscopy (ARPES), scanning single-electron transistor (SET), and scanning nano-scale superconducting quantum interference device (nano-SQUID) measurements. These techniques provide spatially resolved insights into the electronic structure and phase transitions in moiré materials, revealing complex quantum phases that are difficult to access using global probes. The review covers the discovery of flat electronic bands, cascades of electronic transitions, and the identification of symmetry-broken orders in moiré graphene and transition metal dichalcogenides (TMDs). It also discusses the imaging of correlated insulators, electron crystal phases, and charge-ordered phases in TMDs, emphasizing the importance of local probe techniques in understanding the intricate electronic mechanisms in these materials.