A systematic program of investigations of nuclear reactions in laser-produced plasmas is addressed. Such reactions provide an important diagnostic tool for probing the dynamics and thermodynamics in the plasma and understanding laser ion acceleration and neutron production mechanisms. The goal will be to reach the level of knowledge that allows the measurement of fundamental nuclear cross-sections at low and high particle densities. The quantitative measurement of fusion probabilities in hot and dense plasmas will contribute significantly to our comprehension of stellar composition and evolution and will provide important information for development of fusion energy production and applications such as medical isotope production and compact neutron source development. All of these are some of the main goals of the European Extreme Light Infrastructure (ELI), the Shanghai Superintense Ultrafast Laser Facility, and the Station of Extreme Light in China and similar projects in other countries.