1. Introduction.- 2. Ground State.- 2.1 Finite-Size Studies: Rectangular Geometry.- 2.2 Laughlin's Theory.- 2.3 Spherical Geometry.- 2.4 Monte Carlo Results.- 2.5 Reversed Spins in the Ground State.- 2.6 Finite Thickness Correction.- 2.7 Liquid-Solid Transition.- 3. Elementary Excitations.- 3.1 Quasiholes and Quasiparticles.- 3.2 Finite-Size Studies: Rectangular Geometry.- 3.3 Spin-Reversed Quasiparticles.- 3.4 Spherical Geometry.- 3.5 Monte Carlo Results.- 3.6 Experimental Investigations of the Energy Gap.- 3.7 The Hierarchy: Higher Order Fractions.- 4. Collective Modes: Intra-Landau Level.- 4.1 Finite-Size Studies: Spherical Geometry.- 4.2 Rectangular Geometry: Translational Symmetry.- 4.3 Spin Waves.- 4.4 Single Mode Approximation: Magnetorotons.- 5. Collective Modes: Inter-Landau Level.- 5.1 Filled Landau Level.- 5.2 Fractional Filling: Single Mode Approximation.- 5.3 Fractional Filling: Finite-Size Studies.- 6. Further Topics.- 6.1 Effect of Impurities.- 6.2 Higher Landau Levels.- 6.3 Even Denominator Filling Fractions.- 6.4 Half-Filled Landau Level in Multiple Layer Systems.- 7. Open Problems and New Directions.- Appendices.- A The Landau Wave Function in the Symmetric Gauge.- B The Hypernetted-Chain Primer.- C Repetition of the Intra-Mode in the Inter-Mode.- References.