Biased activation of G-protein-coupled receptors (GPCRs) is shifting drug discovery efforts and appears promising for the development of safer drugs. The most effective analgesics to treat acute pain are agonists of the μ opioid receptor (μ-OR), a member of the GPCR superfamily. However, the analgesic use of opioid drugs, such as morphine, is hindered by adverse effects. Only a few μ-OR agonists have been reported to selectively activate the G over β-arrestin signaling pathway, resulting in lower gastrointestinal dysfunction and respiratory suppression. Here, we discuss the strategies that led to the development of biased μ-OR agonists, and potential areas for improvement, with an emphasis on structural aspects of the ligand-receptor recognition process.