Lamar topped everybody with 11 nominations. Swift earned seven, as did producer-singer the Weeknd.
The nominees for album of the year are "1989" (Swift), "Beauty Behind the Madness" (the Weeknd), "To Pimp a Butterfly" (Lamar), "Sound + Color" (Alabama Shakes) and "Traveller" (Chris Stapleton).
The nominees for record of the year are "Blank Space" (Swift), "Uptown Funk!" (Mark Ronson featuring Bruno Mars), "Thinking Out Loud" (Ed Sheeran), "Can't Feel My Face" (the Weeknd) and "Really Love" (D'Angelo and the Vanguard).
The nominees for song of the year -- a songwriter's award -- are "Thinking Out Loud," "Blank Space," "Alright" (performed by Lamar), "Girl Crush" (performed by Little Big Town) and "See You Again" (performed by Wiz Khalifa featuring Charlie Puth).
The nominees for best new artist are Sam Hunt, Meghan Trainor, James Bay, Tori Kelly and Courtney Barnett.
The Grammy attention reaffirms Lamar's importance in the wake of what happened two years ago when his debut, "good kid, m.A.A.d city," came out. He was critically praised then -- and nominated for seven Grammys -- but came up empty for awards.
Now he's topped Eminem as the hip-hop artist with the most nominations in a single year.
"Lamar's achievement is all the more impressive because he did so with 'To Pimp a Butterfly,' a rap album so challenging and musically complex that fans and critics are still chewing on it nearly nine months after its release," the Los Angeles Times' Gerrick D. Kennedy wrote
Recording Academy President/CEO Neil Portnow praised the diversity of the field.
"Artists are pushing boundaries in exciting ways, making it an exceptionally strong year for music," he said in a statement.
The complete list of nominations -- all 83 categories -- is at Grammy.com
Newcomers making waves
Indeed, even genres such as rock and pop, which often lean on old favorites, offered relative newcomers a taste of Grammy glory.
For best rock performance, the nominees included Elle King and Wolf Alice along with Alabama Shakes, Florence + the Machine and Foo Fighters. Best country album nominees are Sam Hunt ("Montevallo"), Kacey Musgraves ("Pageant Material") and Chris Stapleton ("Traveller") along with Ashley Monroe ("The Blade") and Little Big Town ("Pain Killer").
And Seth MacFarlane -- yes, that Seth MacFarlane, the man who created "Family Guy" and "American Dad!" -- earned a slot in the best traditional pop vocal album alongside such Great American Songbook performers as Tony Bennett and Bill Charlap, Josh Groban, Barry Manilow and Bob Dylan. (Yes, that Bob Dylan.)
But where's Adele?
Though Lamar, Swift and the Weeknd led all artists, many in the Twitterverse wondered about the absence of the woman who just sold 4.5 million albums in two weeks: Adele.
Sorry, folks. Adele's album didn't qualify for this year's Grammys because the most recent Grammy year ran from October 2014 to September, and Adele's album didn't come out until November. You can bet she'll be all over the 2017 Grammys.
However, there were some surprising omissions. Billboard editor Joe Levy, talking about the major nominations on CBS, noted that none of Drake's 2015 releases made the cut in the top categories. He also observed that the cast album for the Broadway hit "Hamilton," which uses a variety of musical genres in its score, didn't get an album of the year nomination.
It was a long shot, Levy added. "Hamilton" did pick up a nomination for best musical theater album.
Also, don't weep for Drake. The hip-hop star picked up five nominations, including best rap song ("Energy") and best rap album ("If Youre Reading This Its Too Late").
The Grammys will be held February 15. The show will air from Los Angeles' Staples Center on CBS.