Cells utilize natural supramolecular assemblies to maintain homeostasis and biological functions. Naturally inspired modular assembly of biomaterials are now being exploited for understanding or manipulating cell biology for treatment, diagnosis, and detection of diseases. Supramolecular biomaterials, in particular peptides and oligonucleotides, can be precisely tuned to have diverse structural, mechanical, physicochemical and biological properties. These merits of oligonucleotides and peptides as building blocks have given rise to the evolution of numerous nucleic acid- and peptide-based self-assembling nanomaterials for various medical applications, including drug delivery, tissue engineering, regenerative medicine, and immunotherapy. In this review, we provide an extensive overview of the intracellular delivery approaches using supramolecular self-assembly of DNA, RNA, and peptides. Furthermore, we discuss the current challenges related to subcellular delivery and provide future perspectives of the application of supramolecular biomaterials for intracellular delivery in theranostics.