The theory of the half-filled Landau level explores the behavior of interacting electrons in two dimensions under high magnetic fields, leading to the quantum Hall effect (QHE). At low densities and high magnetic fields, electrons are expected to form a Wigner crystal, but quantum fluctuations cause this to melt into a quantum fluid. The quantum Hall effect exhibits plateaus at filling factors ν = p/q, where q is odd. Laughlin's theory explains these plateaus with incompressible fluids and quasiparticles with fractional charge and statistics. The energy required to create these excitations is related to the system's disorder. While the hierarchy theory predicts smaller gaps for larger denominators, experimental observations suggest a restricted form of filling factors. The nature of the ground state at non-hierarchy fillings remains unclear, with possibilities including phase separation or a pure phase with a Wigner crystal. The paper reviews two approaches: one based on wavefunctions and another field-theoretical, both leading to similar conclusions. The field-theoretical approach uses flux tubes to model anyons, showing that electrons can be represented as fermions with attached flux. The compressible Fermi liquid-like state at ν = 1/q is explained by the formation of quasiparticles with effective mass and statistics. The theory also addresses the dynamics of these quasiparticles, showing their behavior in magnetic fields and the emergence of a Fermi surface. The paper concludes with open questions, including the measurement of effective mass and the validity of Fermi liquid theory. The work is supported by experimental observations confirming the existence of a Fermi surface and agreement with theoretical predictions.The theory of the half-filled Landau level explores the behavior of interacting electrons in two dimensions under high magnetic fields, leading to the quantum Hall effect (QHE). At low densities and high magnetic fields, electrons are expected to form a Wigner crystal, but quantum fluctuations cause this to melt into a quantum fluid. The quantum Hall effect exhibits plateaus at filling factors ν = p/q, where q is odd. Laughlin's theory explains these plateaus with incompressible fluids and quasiparticles with fractional charge and statistics. The energy required to create these excitations is related to the system's disorder. While the hierarchy theory predicts smaller gaps for larger denominators, experimental observations suggest a restricted form of filling factors. The nature of the ground state at non-hierarchy fillings remains unclear, with possibilities including phase separation or a pure phase with a Wigner crystal. The paper reviews two approaches: one based on wavefunctions and another field-theoretical, both leading to similar conclusions. The field-theoretical approach uses flux tubes to model anyons, showing that electrons can be represented as fermions with attached flux. The compressible Fermi liquid-like state at ν = 1/q is explained by the formation of quasiparticles with effective mass and statistics. The theory also addresses the dynamics of these quasiparticles, showing their behavior in magnetic fields and the emergence of a Fermi surface. The paper concludes with open questions, including the measurement of effective mass and the validity of Fermi liquid theory. The work is supported by experimental observations confirming the existence of a Fermi surface and agreement with theoretical predictions.