There's a lot of music out there, but in terms of sticking to well-known musicians, a few come to mind: All the albums for both the Dutch progresive rock band Ayreon and the American neo-soul/hip-hop auteur Janelle Monae are fitted loosely into epic science-fiction narrative continuities. The plot for Ayreon's projects is more well-defined, and deals with a future apocalypse, and the attempt to salvage scraps of humanity through time-travel. Monae's continuity also deals with time-travel, and revolves around the efforts of rebel android/cyborg Cyndi Mayweather to find love, while fighting to free her people from being enslaved by human beings. The most recent of her albums, Dirty Computer, was accompanied by an actual SF film linking the song's music videos into a unified plotline. Ayreon began in the 90s, but continues through today, whereas Monae's solo debut was 2007. A similarly SF-themed narrative can be found in the work of the concept-based "virtual band" Gorillaz, whose entire decades-spanning ouevre can be viewed as the extension of the original conceit.
At a more realist level, the work of indie rock bands Fountains of Wayne and Arcade Fire can be seen as depicting extended storylines about large casts of characters in well-defined suburban milieus.
In terms of single albums, concept albums continue to be made with unified overarching narrative storylines. A notable recent one was American hip-hop collective the Roots' Undun (2011), which depicts the last day in a young man's life in reverse order (it was inspired by the 1999 Prince Paul album A Prince Among Thieves, which has a similar concept).
Pop-music powerhouse Beyonce has also dropped multiple "visual albums," which, like Monae's Dirty Computer, weave together an album's worth of music videos into a single plotline. These include Lemonade, her powerful exploration of the emotional fallout of her husband's affair (Jay-Z's 4:44 was the "he said" side of the conversation), and Black is King, a loose retelling of the narrative of the movie The Lion King with human protagonists.