Drugs based on peptides and proteins (PPs) have been widely used in medicine, beginning with insulin therapy in patients with diabetes mellitus over a century ago. Although the oral route of drug administration is the preferred one by the vast majority of patients and improves compliance, medications of this kind due to their specific chemical structure are typically delivered parenterally, which ensures optimal bioavailability. In order to overcome issues connected with oral absorption of PPs such as their instability depending on digestive enzymes and pH changes in the gastrointestinal (GI) system on the one hand, but also their limited permeability across physiological barriers (mucus and epithelium) on the other hand, scientists have been strenuously searching for novel delivery methods enabling peptide and protein drugs (PPDs) to be administered enterally. These include utilization of different nanoparticles, transport channels, substances enhancing permeation, chemical modifications, hydrogels, microneedles, microemulsion, proteolytic enzyme inhibitors, and cell-penetrating peptides, all of which are extensively discussed in this review. Furthermore, this article highlights oral PP therapeutics both previously used in therapy and currently available on the medical market.