This is an open-label (the researchers and participants know the treatment the participant is receiving) long-term extension study. The study will consist of 2 open-label Phases: 4-week Induction phase (if applicable) and Open-Label Optimization/Maintenance phase (variable). Participants will enter the study Induction Phase from ESKETINTRD3001 (NCT02417064), ESKETINTRD3002 (NCT02418585), ESKETINTRD3003 (NCT02493868), ESKETINTRD3005 (NCT02422186) and ESKETINTRD3006 (US sites only). Participants will enter the study Open-Label Optimization/Maintenance phase from ESKETINTRD3001 (NCT02417064), ESKETINTRD3002 (NCT02418585), ESKETINTRD3003 (NCT02493868) (if appropriate at week 16) or ESKETINTRD3006 (US sites only). In the Open-Label Induction Phase, participants will self-administer flexibly-dosed intranasal esketamine. During first 4 weeks in Optimization/Maintenance Phase responder participants from the induction phase of study 54135419TRD3008, will continue on the same dose of intranasal esketamine from the induction phase and have a weekly intranasal treatment session frequency. Participants entering the optimization/maintenance phase from study ESKETINTRD3005 will also have a weekly intranasal treatment session frequency. However, as the ESKETINTRD3005 intranasal study medication is blinded at the time of entry into the current study, the dose of intranasal esketamine will be administered as outlined in protocol. Participants entering the optimization/maintenance phase from study ESKETINTRD3003 (Direct Entry) or ESKETINTRD3004 who were ongoing in the Optimization, Maintenance, or Optimization/Maintenance phase, respectively, will have the option to have their current intranasal dosing frequency adjusted at the time of entry into 54135419TRD3008 study and should remain on the selected frequency from week 1 to week 4. A one-time dose change will be permitted at study entry. After 4 weeks, intranasal esketamine treatment sessions will be individualized to either once weekly or once every other week at the fixed 2-week interval (based on clinical global impression - severity [CGI-S] performed at that visit), and every 4 weeks for subjects dosed at the 4 week interval. Participants safety will be monitored throughout the study.