This article reviews the effort to understand the physics of high-temperature superconductors from the perspective of doping a Mott insulator. The basic electronic structure of cuprates is discussed, emphasizing strong correlation effects and establishing the model of a doped Mott insulator. Various experiments are reviewed, focusing on the underdoped region where the behavior is most anomalous. The normal state, including the pseudogap phenomenon, is discussed, while the quasiparticles in the superconducting state are well-defined and behave according to theory. The resonating valence bond (RVB) idea is introduced and argued to provide a qualitative account of the data. The importance of phase fluctuation is discussed, leading to a theory of the transition temperature driven by phase fluctuation and thermal excitation of quasiparticles. However, phase fluctuation can only explain the pseudogap phenomenon over a limited temperature range, and additional physics is needed to explain the onset of singlet formation at very high temperatures. The numerical method of projected wavefunction is described as a useful technique to implement strong correlation constraints and make predictions that agree with experiments. The analytic treatment of the $t-J$ model is discussed, aiming to put the RVB idea on a more formal footing. The slave-boson is introduced to enforce the constraint of no double occupation, leading to gauge theories. The historical development of the $U(1)$ and $SU(2)$ formulations of gauge theory is reviewed, along with the role of gauge theory in describing the spin liquid phase of the undoped Mott insulator. The gauge theory formulation of the pseudogap phase is discussed, showing that inclusion of gauge fluctuations provides a reasonable description of the pseudogap phase. The authors argue that $d$-wave superconductivity can be considered as evolving from a stable $U(1)$ spin liquid. The implications of these ideas for the vortex structure and phase diagram of high-$T_c$ cuprates are discussed, and a possible test of the topological structure of the pseudogap phase is proposed.This article reviews the effort to understand the physics of high-temperature superconductors from the perspective of doping a Mott insulator. The basic electronic structure of cuprates is discussed, emphasizing strong correlation effects and establishing the model of a doped Mott insulator. Various experiments are reviewed, focusing on the underdoped region where the behavior is most anomalous. The normal state, including the pseudogap phenomenon, is discussed, while the quasiparticles in the superconducting state are well-defined and behave according to theory. The resonating valence bond (RVB) idea is introduced and argued to provide a qualitative account of the data. The importance of phase fluctuation is discussed, leading to a theory of the transition temperature driven by phase fluctuation and thermal excitation of quasiparticles. However, phase fluctuation can only explain the pseudogap phenomenon over a limited temperature range, and additional physics is needed to explain the onset of singlet formation at very high temperatures. The numerical method of projected wavefunction is described as a useful technique to implement strong correlation constraints and make predictions that agree with experiments. The analytic treatment of the $t-J$ model is discussed, aiming to put the RVB idea on a more formal footing. The slave-boson is introduced to enforce the constraint of no double occupation, leading to gauge theories. The historical development of the $U(1)$ and $SU(2)$ formulations of gauge theory is reviewed, along with the role of gauge theory in describing the spin liquid phase of the undoped Mott insulator. The gauge theory formulation of the pseudogap phase is discussed, showing that inclusion of gauge fluctuations provides a reasonable description of the pseudogap phase. The authors argue that $d$-wave superconductivity can be considered as evolving from a stable $U(1)$ spin liquid. The implications of these ideas for the vortex structure and phase diagram of high-$T_c$ cuprates are discussed, and a possible test of the topological structure of the pseudogap phase is proposed.