Their mom had them acting in one of her later TV shows. I don't know if it is
Life With Lucy, or Here's Lucy, or The Lucy Show. That was sometime in the mid 1960s, I think.
I think that Desi Arnaz Jr stopped acting in his moms TV shows before his sister Lucie did .
I best recall that Lucie Arnaz and Desi Arnaz Jr actually played the son and daughter of their mother in her TV show.

But Lucie Arnaz did at least one Broadway musical sometime in the 70s.
I have not really heard of her or her brother since.

What are they up to in the world of show biz, or are they retired?

Desi Jr popped up in a few movies in the 70's and 80's (Most notably Joyride '77 with Robert Carridine and Melanie Griffith, then in 1983 appeared alongside the likes of Christopher Lee, Peter Cushing, Vincent Price, and John Carradine in House Of Long Shadows), and starred in the short lived 80's TV series Automan.
His last film appearance was 1992 playing his dad in The Mambo Kings.

Lucie Arnaz http://www.imdb.com/name/nm0036109/
Desi Arnaz Jr. http://www.imdb.com/name/nm0036107/

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Their mom had them acting in one of her later TV shows. I don't know if it is
Life With Lucy, or Here's Lucy, or The Lucy Show. That was sometime in the mid 1960s, I think.
I think that Desi Arnaz Jr stopped acting in his moms TV shows before his sister Lucie did .
I best recall that Lucie Arnaz and Desi Arnaz Jr actually played the son and daughter of their mother in her TV show.

But Lucie Arnaz did at least one Broadway musical sometime in the 70s.
I have not really heard of her or her brother since.

What are they up to in the world of show biz, or are they retired?

Lucie Arnaz co-starred with her mother for six seasons on Here's Lucy: 1968-1974.

Desi Arnaz, Jr. only hung around for the first three seasons because he got busy with movie offers. His mother set him free to pursue his movie career which kept him active for awhile. But by the '80s, and after starring on the cult hit Automan, he got burnt out from the Hollywood life, and in the early or mid-80s, with both his parents by his side for support, he went into drug rehab. By the end of the decade he had decided to leave Hollywood behind and move to Boulder City, Nevada for a quiet life with his wife, Amy, where she opened a dance school. He made a rare exception in 1992 and stepped back into Hollywood to play his father in the Antonio Banderas film, The Mambo Kings. But mostly, he stayed in Boulder City, Nevada, supporting his wife's venture. They were married for more than 27 years before she passed away in 2015. Desi still loves music occasionally performs in public, but mostly lives quietly.

https://www.youtube.com/watch?v=yXCgIx2Bycs

I'm sure the last couple of years have been challenging without his beloved Amy.

As for Lucie Arnaz, she has never stopped. She has the luxury of doing what she loves, which is Broadway and cabaret singing. She's had a very extensive career on Broadway and the London stage, as well as touring. Most recently, last year, she took over the Andrea Martin starring role for the national tour of Pippen. After leaving Pippen, she jumped back into her cabaret shows, which gives her the flexibility to work when she wants so she can spend more time with her husband of 37 years, Laurence Luckinbill. They now live in Palm Springs.

Here's a taste of her cabaret performances:

This is a tribute to her mother's Wildcat. You must listen to the end. She's quite the singer: https://www.youtube.com/watch?v=MHOtxWyYc7s

Here's something more recent, honoring her father: https://www.youtube.com/watch?v=yJNzvphx6iU

Last year, she went on a Turner Classic Movies cruise honoring her mother: https://www.youtube.com/watch?v=O8c_mrCuWuI

Her husband is in his 80s now, so she announced after Pippen that she would no longer be working full time so as to spend more time with him. But she does still keep some shows on her calendar. You can find news about upcoming performances (she's done one in SoCal within the last week or two) here: http://www.luciearnaz.com/


Here's a fairly recent interview: https://www.youtube.com/watch?v=HNToJIVO0PE

And here's one Lucie and Desi Jr. did together when they were both about to perform together in honor of the father: https://www.youtube.com/watch?v=nBD82B5p_Fk

Are either of them grandparents?

Are either of them grandparents?

Desi Jr. has been a grandparent for a long time. Lucie doesn't have any grandchildren yet, but her daughter, Kate, was married last summer so that will probably be happening before too long. https://vimeo.com/173842895

Lucie and her husband, Lawrence Luckinbill, moved to Palm Springs a few years ago. They lived in New York for many years. I know Desi Jr.'s wife died of cancer a few years ago and he lives just outside of Las Vegas.