Scottish champions Celtic, the relative minnows of the draw, were pitted against Italian giants Juventus.
The first legs will take place on February 12, 13, 19 and 20 and the return games on March 5, 6, 12 and 13.
Three-times winners United, whose last competitive clash against Real was a thrilling Champions League knockout clash in 2003, will be reunited with their former winger Cristiano Ronaldo.
The 'other' Ronaldo famously hit a hat-trick in their last competitive clash, a 4-3 win for United that saw nine-times European champions Real go through 6-5 on aggregate.
Arsenal, who have never won Europe's premier club competition, face one of the few sides to have the upper hand on them in European football, with four-times champions and in-form Bundesliga leaders Bayern the prize for Arsene Wenger's side.
Bayern actually lost 1-0 to Arsenal in their last encounter in 2005.Thierry Henry's strike the difference, but the German giants went through on aggregate.
Favourites Barcelona must negotiate former winners AC Milan, who have fallen on hard times recently.
Barcelona vice-president Josep Maria Bartomeu told Spanish broadcaster Canal Plus: "Milan are one of Europe's greats and a club that commands a huge amount of respect.
"They are not going well in their league right now but they have some excellent players. They are not currently at their best but football changes from one day to the next."
Other clashes see Turkish side Galatasaray face Germany's Schalke 04, while 2009 UEFA Cup winners Shakhtar Donetsk host Bundesliga champions Borussia Dortmund.
Qatari-owned Paris Saint-Germain visit Spain's Valencia in their first leg, while 1986 and 2004 winners Porto host another oil-rich club in Malaga.
Champions League last 16 draw
Galatasaray v FC Schalke
Celtic v Juventus
Arsenal v Bayern Munich
Shakhtar Donetsk v Borussia Dortmund
AC Milan v Barcelona
Real Madrid v Manchester United
Valencia v Paris Saint-Germain
Porto v Malaga
First leg
February 12
Celtic v Juventus
Valencia v Paris St Germain
February 13
Shakhtar Donetsk v Borussia Dortmund
Real Madrid v Manchester United
February 19
Porto v Malaga
Arsenal v Bayern Munich
February 20
Galatasaray v Schalke 04
AC Milan v Barcelona
Second leg
March 5
Manchester United v Real Madrid
Borussia Dortmund v Shakhtar Donetsk
March 6
Paris St Germain v Valencia
Juventus v Celtic
March 12
Schalke 04 v Galatasaray
Barcelona v AC Milan
March 13
Malaga v Porto
Bayern Munich v Arsenal
