Click chemistry has become one of the most powerful construction tools in the field of organic chemistry, materials science, and polymer science, as it offers hassle-free platforms for the high-yielding synthesis of novel materials and easy functionalization strategies. The absence of harsh reaction conditions or complicated workup procedures allowed the rapid development of novel biofunctional polymeric materials, such as biopolymers, tailor-made polymer surfaces, stimulus-responsive polymers, etc. In this review, we discuss various types of click reactions─including azide-alkyne cycloadditions, nucleophilic and radical thiol click reactions, a range of cycloadditions (Diels-Alder, tetrazole, nitrile oxide, etc.), sulfur fluoride exchange (SuFEx) click reaction, and oxime-hydrazone click reactions─and their use for the formation and study of biofunctional polymers. Following that, we discuss state-of-the-art biological applications of "click"-biofunctionalized polymers, including both passive applications (e.g., biosensing and bioimaging) and "active" ones that aim to direct changes in biosystems, e.g., for drug delivery, antiviral action, and tissue engineering. In conclusion, we have outlined future directions and existing challenges of click-based polymers for medicinal chemistry and clinical applications.