This lecture discusses various types of metal-insulator transitions, including Anderson transitions in non-crystalline systems and Mott transitions in crystalline systems. In non-crystalline systems, an Anderson transition occurs due to disorder, leading to localization of electrons. In crystalline systems, band-crossing and Mott transitions are discussed, both involving discontinuities in carrier number when electron-electron interactions are considered. However, sufficient disorder can remove this discontinuity. Examples of different materials are given, including Ti₂O₃, VO₂, V₂O₃, NiS₂, and NiS, each with its own transition mechanism. A new suggestion is made for Ti₂O₃, where the band gap approaches the critical value E₀. The Verwey transition in Fe₃O₄ is also mentioned. In non-crystalline systems, the Anderson transition is characterized by a change in conductivity with temperature, showing variable-range hopping at low temperatures and exponential behavior at high temperatures. The transition is associated with a critical value of the mobility edge, and the minimum conductivity σ_min is discussed. The theory for these transitions involves non-interacting electrons, though interactions may affect magnetic properties. In crystalline systems, band-crossing transitions occur when the band gap changes, leading to a discontinuous change in carrier number and activation energy. Mott transitions involve electron-electron interactions and can result in a discontinuous change in carrier number and activation energy. These transitions are often associated with changes in magnetic properties and electronic specific heat. The lecture also discusses the role of disorder in these transitions, noting that quenched alloys may exhibit Anderson-type transitions. The transition in Ti₂O₃ is particularly interesting, as it involves a change in the band gap and the effective mass of electrons. The transition in VO₂ is attributed to changes in the degree of s-p hybridization, leading to a change in the Hubbard U parameter. The lecture concludes with a discussion of the Verwey transition in Fe₃O₄, where the material transitions from a non-metallic to a metallic state at low temperatures, involving a reordering of Fe³+ ions. The transition is associated with a narrow d band and is more likely to occur in oxides than in sulphides. The behavior of Ti₄O₇ is also discussed, showing a transition involving mobile polarons. The lecture highlights the complexity of these transitions and the need for further theoretical and experimental investigation.This lecture discusses various types of metal-insulator transitions, including Anderson transitions in non-crystalline systems and Mott transitions in crystalline systems. In non-crystalline systems, an Anderson transition occurs due to disorder, leading to localization of electrons. In crystalline systems, band-crossing and Mott transitions are discussed, both involving discontinuities in carrier number when electron-electron interactions are considered. However, sufficient disorder can remove this discontinuity. Examples of different materials are given, including Ti₂O₃, VO₂, V₂O₃, NiS₂, and NiS, each with its own transition mechanism. A new suggestion is made for Ti₂O₃, where the band gap approaches the critical value E₀. The Verwey transition in Fe₃O₄ is also mentioned. In non-crystalline systems, the Anderson transition is characterized by a change in conductivity with temperature, showing variable-range hopping at low temperatures and exponential behavior at high temperatures. The transition is associated with a critical value of the mobility edge, and the minimum conductivity σ_min is discussed. The theory for these transitions involves non-interacting electrons, though interactions may affect magnetic properties. In crystalline systems, band-crossing transitions occur when the band gap changes, leading to a discontinuous change in carrier number and activation energy. Mott transitions involve electron-electron interactions and can result in a discontinuous change in carrier number and activation energy. These transitions are often associated with changes in magnetic properties and electronic specific heat. The lecture also discusses the role of disorder in these transitions, noting that quenched alloys may exhibit Anderson-type transitions. The transition in Ti₂O₃ is particularly interesting, as it involves a change in the band gap and the effective mass of electrons. The transition in VO₂ is attributed to changes in the degree of s-p hybridization, leading to a change in the Hubbard U parameter. The lecture concludes with a discussion of the Verwey transition in Fe₃O₄, where the material transitions from a non-metallic to a metallic state at low temperatures, involving a reordering of Fe³+ ions. The transition is associated with a narrow d band and is more likely to occur in oxides than in sulphides. The behavior of Ti₄O₇ is also discussed, showing a transition involving mobile polarons. The lecture highlights the complexity of these transitions and the need for further theoretical and experimental investigation.