The study of manganites, which exhibit the "Colossal" Magnetoresistance (CMR) effect, is a key area in the research of strongly correlated electrons. Recent theoretical studies, guided by computational and mean-field models, have advanced understanding of these materials' properties. These studies suggest that manganite ground states are intrinsically inhomogeneous due to phase separation tendencies, involving ferromagnetic metallic and antiferromagnetic charge/orbital ordered insulating domains. Calculations of resistivity vs. temperature using mixed states agree with experiments. Phase separation has two origins: electronic phase separation between different density phases and disorder-induced phase separation with percolative characteristics. Coexisting clusters can be as large as micrometers. Theoretical predictions align with experimental results, and mixed-phase states' phenomenology is independent of model details. However, clarifying electronic properties based on microscopic Hamiltonians is crucial. Open questions remain, and mixed-phase phenomenology may appear in various compounds, including ruthenates and diluted magnetic semiconductors. Manganites' diverse physical phenomena make them important for correlated electron research. Manganites, first studied by Jonker and van Santen (1950), are manganese oxides with the formula T₁₋ₓDₓMnO₃. Later studies by Wollan and Koehler (1955) characterized antiferromagnetic and ferromagnetic phases, identifying a CE-state. Theoretical work explained ferromagnetism via "double-exchange" (DE). The 1990s saw renewed interest with large magnetoresistance (MR) effects in compounds like Nd₀.₅Pb₀.₅MnO₃ and La₂/₃Ba₁/₃MnOₓ. The CMR effect, with MR ratios up to 127,000%, was observed in La₀.₆₇Ca₀.₃₃MnOₓ. This led to renewed interest in manganites. Early work focused on x=0.3 doping, but recent attention has shifted to other densities. Understanding CMR requires knowledge of both ferromagnetic and competing phases. Manganites are classified into large, intermediate, and small bandwidth compounds. Large-bandwidth manganites like La₁₋ₓSrₓMnO₃ have high Curie temperatures and complex behavior near x=1/8. Intermediate-bandwidth manganites like La₁₋ₓCaₓMnO₃ exhibit large CMR effects due to charge ordering. Low-bandwidth manganites like Pr₁₋ₓCaₓMnO₃ have stable charge-ordered states. Other perovskite manganites, such as Nd₁₋ₓSrₓMnO₃, also show CO phases. Double-layer and single-layer compounds exhibit different behaviors, with single-layer manganites showingThe study of manganites, which exhibit the "Colossal" Magnetoresistance (CMR) effect, is a key area in the research of strongly correlated electrons. Recent theoretical studies, guided by computational and mean-field models, have advanced understanding of these materials' properties. These studies suggest that manganite ground states are intrinsically inhomogeneous due to phase separation tendencies, involving ferromagnetic metallic and antiferromagnetic charge/orbital ordered insulating domains. Calculations of resistivity vs. temperature using mixed states agree with experiments. Phase separation has two origins: electronic phase separation between different density phases and disorder-induced phase separation with percolative characteristics. Coexisting clusters can be as large as micrometers. Theoretical predictions align with experimental results, and mixed-phase states' phenomenology is independent of model details. However, clarifying electronic properties based on microscopic Hamiltonians is crucial. Open questions remain, and mixed-phase phenomenology may appear in various compounds, including ruthenates and diluted magnetic semiconductors. Manganites' diverse physical phenomena make them important for correlated electron research. Manganites, first studied by Jonker and van Santen (1950), are manganese oxides with the formula T₁₋ₓDₓMnO₃. Later studies by Wollan and Koehler (1955) characterized antiferromagnetic and ferromagnetic phases, identifying a CE-state. Theoretical work explained ferromagnetism via "double-exchange" (DE). The 1990s saw renewed interest with large magnetoresistance (MR) effects in compounds like Nd₀.₅Pb₀.₅MnO₃ and La₂/₃Ba₁/₃MnOₓ. The CMR effect, with MR ratios up to 127,000%, was observed in La₀.₆₇Ca₀.₃₃MnOₓ. This led to renewed interest in manganites. Early work focused on x=0.3 doping, but recent attention has shifted to other densities. Understanding CMR requires knowledge of both ferromagnetic and competing phases. Manganites are classified into large, intermediate, and small bandwidth compounds. Large-bandwidth manganites like La₁₋ₓSrₓMnO₃ have high Curie temperatures and complex behavior near x=1/8. Intermediate-bandwidth manganites like La₁₋ₓCaₓMnO₃ exhibit large CMR effects due to charge ordering. Low-bandwidth manganites like Pr₁₋ₓCaₓMnO₃ have stable charge-ordered states. Other perovskite manganites, such as Nd₁₋ₓSrₓMnO₃, also show CO phases. Double-layer and single-layer compounds exhibit different behaviors, with single-layer manganites showing