Nucleons (protons and neutrons) are the building blocks of atomic nuclei, and are responsible for more than 99% of the visible matter in the universe. Despite decades of efforts in studying its internal structure, there are still a number of puzzles surrounding the proton such as its spin, and charge radius. Accurate knowledge about the proton charge radius is not only essential for understanding how quantum chromodynamics (QCD) works in the non-perturbative region, but also important for bound state quantum electrodynamics (QED) calculations of atomic energy levels. It also has an impact on the Rydberg constant, one of the most precisely measured fundamental constants in nature. This article reviews the latest situation concerning the proton charge radius in light of the new experimental results from both atomic hydrogen spectroscopy and electron scattering measurements, with particular focus on the latter. We also present the related theoretical developments and backgrounds concerning the determination of the proton charge radius using different experimental techniques. We discuss upcoming experiments, and briefly mention the deuteron charge radius puzzle at the end. CONTENTS I. Introduction 1 II. The Proton Charge Radius Puzzle 3 A. Before the millennium 3 B. After the millennium 3 III. Elastic Electron-Proton Scattering 3 A. Introduction to Electron-Proton Scattering and Proton Electromagnetic Form Factors 3 B. Three-dimensional parton distributions 5 C. The nucleon transverse charge densities 7 D. GPDs and quark densities in longitudinal momentum and transverse position in a proton 10 E. Radii of quark distributions in a proton 12 F. Unpolarized electron-proton elastic scattering 16 G. Double-Polarization Elastic Electron-Proton Scattering 17 1. Spin-Dependent Asymmetry from p( e, e )p 17 2. Recoil Proton Polarization from p( e, e p) 18 H. Two-Photon-Exchange Contribution to Electron-Proton Scattering 18 I. Radiative Corrections in Electron Scattering 20 J. The Extraction of the Proton Charge Radius from Proton Electric Form Factor 21 IV. Atomic Hydrogen Spectroscopy 21 V. Modern lepton scattering experiments 24 A. Mainz 2010 24 B. JLab recoil polarization experiment 24 C. Mainz ISR measurements 24 D. The PRad experiment at JLab 25 E. Proton charge radius from modern analyses of proton electric form factor data 29 VI. Modern spectroscopic measurements 31 A. Muonic hydrogen spectroscopic experiments 31 B. Ordinary Hydrogen spectroscopic experiments 33 VII. Ongoing and upcoming experiments 35 A. The MUSE experiment at PSI 35 B. The COMPASS++/AMBER Experiment at CERN 35 C. The PRad-II experiment at Jefferson Lab 36 D. Future electron scattering experiments at Mainz 38 E. The ULQ 2 experiment at Tohoku University 39 VIII. The deuteron charge radius 39 IX. Conclusions 42 X. Acknowledgments 42 References 43