This article presents an overview of ongoing research at Oak Ridge National Laboratory on liquid metal (LM) magnetohydrodynamic (MHD) flows under conditions relevant to fusion breeding/cooling applications, such as blankets and plasma-facing components for a future U.S. fusion pilot plant. The research is conducted within four fusion projects in which significant effort is taken to characterize closed and open-surface MHD flows of electrically conducting LM breeder/coolant (liquid lithium Li or eutectic lead-lithium PbLi alloy) and associated heat and mass transfer processes under various fusion reactor operation scenarios and conditions. This includes strong magnetic fields, stationary and transient plasmas, and high thermal loads. A special effort is dedicated to the development of a new experimental facility called Corrosion Upgraded PbLi Experiment (CoUPLE) and the associated experimental program for studying corrosion processes of structural and functional fusion materials in flowing PbLi, in a strong magnetic field.