Holders Celtic have been drawn at home to Championship outfit Greenock Morton in the Scottish Cup quarter-finals.
The other surviving second-tier side, Falkirk, also face a tough draw away to Rangers, who thumped League One side Ayr United 6-1 on Sunday.
There are two all-Premiership ties following Sunday's draw.
Last year's runners-up, Aberdeen, who defeated Dundee United 4-2 on Sunday, host Kilmarnock, while Hearts are away to Motherwell.
The Dons last played Killie in the Scottish Cup in 1984 and went through in a replay at Rugby Park before the Pittodrie side went on to lift the trophy.
They have never lost in seven previous Scottish Cup meetings with the Ayrshire side, who has lost 14 of their last 15 meetings in the Premiership.
But Kilmarnock have improved since Steve Clarke took over as manager in October and Dons manager Derek McInnes realises the threat from a side whose only defeat in their last 10 outings was at Pittodrie on league duty in January.
"I'm really pleased with the fact we are at home again," McInnes told BBC Scotland.
"We are obviously up against a tough opponent. Steve's got them going well as we know and they are getting a lot of credit - and rightly so.
"And, although we've beaten them a couple of times, they have been tight games and we've had to fight hard for them and had to bring quality to separate the teams.
"It is a quarter-final and both teams will see it as an opportunity to get into the semi-final.
"It is an exciting tie for both teams."
Although they have won the League Cup, the Dons have yet to lift the Scottish Cup under McInnes and he admits to some frustration at that.
"I think we've had five semi-finals in four years and three cup finals," he said of both competitions. "At least we are putting ourselves in position to win them.
"I feel we have gained good experience, although it is a different squad now, and hopefully we can go the full way.
"I think the team is good enough and we are now expected to be competing at the end of tournaments and we will be disappointed if we don't."
Scottish Cup quarter-final draw
Celtic v Greenock Morton
Aberdeen v Kilmarnock
Motherwell v Heart of Midlothian
Rangers v Falkirk
Ties to played on weekend 3/4 March