Ribonucleoprotein (RNP) granules transport, store, or degrade messenger RNAs, thereby indirectly regulating protein synthesis. Normally, RNP granules are highly dynamic compartments. However, because of aging or severe environmental stress, RNP granules, in particular stress granules (SGs), convert into solid, aggregate-like inclusions. There is increasing evidence that such RNA-protein inclusions are associated with several age-related neurodegenerative diseases, such as amyotrophic lateral sclerosis (ALS), fronto-temporal dementia (FTD) and Alzheimer’s disease (AD). Thus, understanding what triggers the conversion of RNP granules into aggregates and identifying the cellular players that control RNP granules will be critical to develop treatments for these diseases. In this review article, we discuss recent insight into RNP and SG formation. More specifically, we examine the evidence for liquid-liquid phase separation (LLPS) as an organizing principle of RNP granules and the role of aggregation-prone RNA-binding proteins (RBPs) in this process. We further discuss recent findings that liquid-like SGs can sequester misfolded proteins, which promote an aberrant conversion of liquid SGs into solid aggregates. Importantly, very recent studies show that a specific protein quality control (PQC) process prevents the accumulation of misfolding-prone proteins in SGs and, by doing so, maintains the dynamic state of SGs. This quality control process has been referred to as granulostasis and it relies on the specific action of the HSPB8-BAG3-HSP70 complex. Additional players such as p97/valosin containing protein (VCP) and other molecular chaperones (e.g., HSPB1) participate, directly or indirectly, in granulostasis, and ensure the timely elimination of defective ribosomal products and other misfolded proteins from SGs. Finally, we discuss recent findings that, in the stress recovery phase, SGs are preferentially disassembled with the assistance of chaperones, and we discuss evidence for a back-up system that targets aberrant SGs to the aggresome for autophagy-mediated clearance. Altogether the findings discussed here provide evidence for an intricate network of interactions between RNP granules and various components of the PQC machinery. Molecular chaperones in particular are emerging as key players that control the composition and dynamics of RNP granules, which may be important to protect against age-related diseases.