Nanoparticles find many uses in medicine and biomedical technology. Such applications imply that they must be colloidally stable and do not interact with proteins in the blood or blood serum. A nanoparticle put into the blood will instantaneously be covered by a protein corona that compromises the function of the nanoparticle core, changes the effective size of the nanoparticle, and determines its biological fate. Strategies developed to gain control over nanoparticles in biological fluids, particular in blood, heavily rely on creating a hydrated polymer shell that sterically and osmotically prevents a protein corona from forming. In this tutorial review, we provide an overview of factors that affect the formation of the protein corona in blood and how to prevent it forming. We focus on describing the latest advances in our understanding of how small core-shell nanoparticles (core diameter 4-20 nm in diameter) with a shell of densely grafted polymer chains, a so-called polymer brush, interact with proteins and cells in vitro. Such nanoparticles are among the most well-defined and well-characterized colloids used for biomedical applications, from which an improved understanding of how nanoparticle architecture influences their biological fate can be obtained in detail.