With the new season of the successful Encores! Off-Center, which celebrates off-Broadway musicals of the past, starting next week, it brings up the question: Does a genuine distinction still exist between a Broadway musical and an off-Broadway musical?

Major not-for-profit off-Broadway companies (like the Public Theater and Second Stage, in particular) continue to produce new musicals every season. But if the show is successful enough to warrant a longer run, it will usually transfer to Broadway, as it is no longer a viable option for a musical to transfer to one of the few remaining commercial off-Broadway venues.

Off-Broadway musicals do tend to be smaller in size and more original and adventurous in nature. But with the exception of “Little Shop of Horrors,” the most successful off-Broadway musicals of the past 50 years are now thought of as Broadway musicals, from “Hair” and “A Chorus Line” to “Rent” and “Hamilton.”

As such, I was somewhat skeptical when City Center first announced that it would create an offshoot of its much beloved Encores! series (which celebrates rarely seen Broadway musicals) that would give the same treatment to off-Broadway musicals of the past.

Its first three seasons included concert stagings of “The Cradle Will Rock,” “Violet” (with Sutton Foster, which transferred to Broadway), “Tick, Tick ... Boom!,” “The Wild Party” and “Little Shop” (with Jake Gyllenhaal and original star Ellen Greene).

The fourth season begins next week with a short run of “Runaways,” a 1978 revue about emotionally troubled runaway teens that premiered at the Public Theater and had a short run on Broadway. Elizabeth Swados, who wrote the music, lyrics and book, died a few months ago. The cast of the Off-Center production consists primarily of local high school students.

On July 16, Sutton Foster and Jonathan Groff will join “Fun Home” composer Jeanine Tesori (who is about to retire as the Off-Center artistic director) for the “Off-Center Jamboree!,” a concert of songs from off-Broadway musicals.

The season will finish at the end of July with “Kurt Vonnegut’s God Bless You, Mr. Rosewater,” the 1979 debut musical of Alan Menken and Howard Ashman, based on Vonnegut’s 1965 novel about a millionaire who is accused of insanity when he tries to give away his fortune. The cast includes James Earl Jones, Skylar Astin and Santino Fontana.

Personally, these are just a few off-Broadway musicals I’d like to see produced by Off-Center in future summers: “The Human Comedy,” “Floyd Collins,” “Bat Boy,” “Songs for a New World,” “Lucky Stiff,” “Dogfight,” “Heathers,” “Caroline, or Change,” “Reefer Madness” and “Bright Lights, Big City.”