Laser fusion, likely the ultimate solution to the crisis of human energy, is highly valued by the international community and has always been the focus of international research. It turns out that the biggest scientific obstacle of laser fusion is the effective control of the high-energy-density nonlinear flows during implosions. The research of high-energy-density nonlinear flows covers many different fields, such as high-energy-density physics, plasma physics, fluid mechanics, computing science, strong impact physics, and high pressure atomic physics. Meanwhile, the capability of multi-material and multi-scale numerical simulations as well as large laser facility with high output power is also needed. As an emerging research field, it is full of all kinds of novel phenomena to be explored. In addition, hydrodynamic instabilities and the subsequent turbulent mixing in high-energy-density flows, are also important processes in astrophysical phenomena (e.g., galaxy collision and merging, stellar evolution, formation of protostars and supernova explosion) and involve with the core content of astrophysics. This paper reviews, firstly the status and progress, as well as the challenges and opportunities of high-energy-density nonlinear flows research. Secondly, it introduces hydrodynamic instabilities during implosions in central ignition laser fusion, among which, key factors related to the bottleneck of implosion performance of the National Ignition Facility (NIF) in the United States are condensed. Next, it summarizes the development of hydrodynamic instability experiments in laser fusion abroad. Finally, it lists some key achievements on the fundamental issues of hydrodynamic instabilities by the laser fusion implosion physics team in China over the last three years. This team has been engaged in the research and control of nonlinear flows in laser fusion implosions, as well as the research and design of target physics. A lot of improvements have been made in recent years on the theoretical analysis and numerical simulation of outstanding issues for hydrodynamic instabilities in laser fusion implosions, and the design and analysis of experiments on large lasers, which greatly promoted the development of this research direction in China.