Update, 12:15 PM EST, January 31: Replay functionality has now been fully restored, as Epic Games have completed the necessary maintenance.
Lights💡— Fortnite (@FortniteGame) January 31, 2019
We’ve re-enabled the ability to watch replays. Load up your previous games and relive your best plays!
The replay functionality was one of the most requested features when the battle royale mode first rose to popularity, but it can put serious strain on both your hardware and the Fortnite servers.
As such, it requires constant maintenance, and Epic Games selected January 30 as the day to perform some of the necessary upkeep.
Although they haven't gone into specifics about the issue which was occurring, they explained that there was a problem with the feature which needed some attention.
Some players noticed that Marshmello's music - which will be played at the Festivus event on February 2 - was audible while in replay mode, so this could be why Epic are being secretive about the issue.
While they investigated, the ability to view replays was been turned off - but it's not all bad news.
If you do manage to have a remarkable game you want to watch back later, Epic has confirmed that any matches played during the downtime will be saved and watchable now that everything is back up and running.
We’ve disabled the ability to view replays while we investigate an issue.— Fortnite (@FortniteGame) January 30, 2019
All replays will still be saved during this time and watchable once this feature has been re-enabled.
Of course, you can also always use your devices' own capturing equipment, if you are playing on PS4, Xbox One or PC, you will be able to record your gameplay with the inbuilt technology - although it's not as powerful as the full functionality of replay mode of course.