Elastic moduli and internal frictions are fundamental properties of solid materials, and they are usually fairly sensitive to temperature, microstructure, deformation, and external fields. Measurement of elastic moduli and internal frictions is very important and useful in both academics and engineering. In this work, we first briefly reviewed the measurement methods of elastic moduli and internal frictions over the past 100 years and the related applications. Next, a promising measurement method called modified piezoelectric ultrasonic composite oscillator technique (M-PUCOT) was presented which can measure Young's modulus, shear modulus, and the related internal frictions accurately and quickly. Then, some typical measurement results using M-PUCOT on metals and phase transition materials were presented from room temperature up to 500 °C. The M-PUCOT measurement on rocks after different levels of compression was also conducted to predict the damage before fracture. The perspective and challenge of the M-PUCOT method as the indicator of heat treatment, mechanical fatigue, and early damage of metals and ceramics were further presented and discussed. It is expected that, in the near future, the measurement of elastic moduli and internal friction will be as convenient as that of electric conductivity, permittivity, etc. and can turn to be a multi-functional tool for many advanced applications.