FUS and TDP-43, two RNA-binding proteins from the heterogeneous nuclear ribonucleoprotein family, have gained significant attention in the field of neurodegenerative diseases due to their association with amyotrophic lateral sclerosis (ALS) and frontotemporal degeneration (FTD). They possess folded domains for binding ATP and various nucleic acids including DNA and RNA, as well as substantial intrinsically disordered regions (IDRs) including prion-like domains (PLDs) and RG-/RGG-rich regions. They play vital roles in various cellular processes, including transcription, splicing, microRNA maturation, RNA stability and transport and DNA repair. In particular, they are key components for forming ribonucleoprotein granules and stress granules (SGs) through homotypic or heterotypic liquid-liquid phase separation (LLPS). Strikingly, liquid-like droplets formed by FUS and TDP-43 may undergo aging to transform into less dynamic assemblies such as hydrogels, inclusions, and amyloid fibrils, which are the pathological hallmarks of ALS and FTD. This review aims to synthesize and consolidate the biophysical knowledge of the sequences, structures, stability, dynamics, and inter-domain interactions of FUS and TDP-43 domains, so as to shed light on the molecular mechanisms underlying their liquid-liquid phase separation (LLPS) and amyloidosis. The review further delves into the mechanisms through which ALS-causing mutants of the well-folded hPFN1 disrupt the dynamics of LLPS of FUS prion-like domain, providing key insights into a potential mechanism for misfolding/aggregation-prone proteins to cause neurodegenerative diseases and aging by gain of functions. With better understanding of different biophysical aspects of FUS and TDP-43, the ultimate goal is to develop drugs targeting LLPS and amyloidosis, which could mediate protein homeostasis within cells and lead to new treatments for currently intractable diseases, particularly neurodegenerative diseases such as ALS, FTD and aging. However, the study of membrane-less organelles and condensates is still in its infancy and therefore the review also highlights key questions that require future investigation.