Seismic hazard analysis focuses on the probability analysis of large earthquakes, whereas small and moderate-magnitude earthquakes occurred much more often and can be used to estimate earthquake source properties in a more routine fashion. In this talk I will present some recent developments in understanding the physics of earthquakes using small and moderate-magnitude events. Instead of relying on absolute earthquake magnitudes, we have developed a relative magnitude method to quantify the differences between earthquake magnitudes. The magnitude-frequency distribution based on relative magnitudes produces more robust estimates of b-values that characterize the relative abundance of large earthquakes to small earthquakes and thus the potential of increasing earthquake hazard in the region. Another fundamental source parameter is stress drop, i.e., the difference of fault shear stresses before and after an earthquake, and higher stress drops usually lead to more high-frequency ground motions. It has been proposed that injection-induced earthquakes have lower stress drop due to high pore fluid pressure in the source region; however, our recent analysis showed that stress drops of moderate-magnitude induced earthquakes are similar to those of tectonic earthquakes in the central US. We further analyzed stress drops of very small (M -3.9 to -3.1) induced earthquakes during a fluid injection experiment in France and found very low stress drops, indicating a dependency of stress drop on crustal shear strength. In the end, I will also demonstrate that the source spectra of small earthquakes used to estimate stress drop have amplified high-frequency energy at directions close to fault strike. Such high-frequency amplification may be caused by damaged rocks around faults and can greatly contribute to strong ground motions during large earthquakes.