The first leg will be played away in Bosnia-Herzegovina, while the second leg will take place in Aviva Stadium. Dates will be confirmed later today.
Martin O'Neill's team qualified for the play-offs after finishing third in Group D and they were unseeded for the draw which took place in Nyon, Switzerland on Sunday morning.
The first leg will take place in Bosnia-Herzegovina on Friday, November 13, while the second leg will be hosted in Aviva Stadium on Monday, November 16 - both games will kick off at 19:45.
Ireland manager O'Neill was present at the draw in Nyon, Switzerland on Sunday morning and gave his reaction after his team - who were unseeded - were drawn against Bosnia-Herzegovina.
"If there is any advantage to be had, then we have the second game at home in Dublin. But that said we need to perform out in Bosnia to make sure that the second game means something. That will not be easy but the draw is there and we have a chance," said O'Neill.
Ticket details for the home leg will be announced in due course.
UEFA 2016 European Championship Play-Off
Bosnia-Herzegovina v Republic of Ireland, First Leg, Friday, November 13, KO 19:45
Republic of Ireland v Bosnia-Herzegovina, Second Leg, Monday, November 16, KO 19:45