When They Might Be Giants first appeared on the musical landscape 25 years ago, to call them outliers would have been wholly insufficient. Armed with their trademark literary preoccupations, indelible melodies and deranged dada-esque touches, the band’s strangeness was nearly Martian in scale. In terms of commercial prospects, their timing could scarcely have been worse.

The music industry in 1986 would hardly be recognizable today. During that era of good feeling and guitar-shaped swimming pools, producing the next pop star remained a billion-dollar proposition. The means of production and distribution were still controlled by a coterie of powerful major labels not noted for their discerning taste. Billboard’s Top 5 albums from November of that year were largely a mirror image of the pumped-up, “Morning in America” optimism that had infected so much of the nation during Ronald Reagan’s second term. There was “Third Stage” by Boston (No. 1), “Fore!” by Huey Lewis and the News (No. 3) and (of course) the soundtrack for “Top Gun” (No. 5).

“God knows, it hasn’t been easy,” reflects John Linnell, on his professional odyssey as one-half of the musical institution that has become They Might Be Giants (the other half being his high school classmate John Flansburgh). “But I think we always felt it wasn’t about commercial success for us. The prospect of not being successful in the future didn’t deter us, so that’s probably why we imagined doing this into middle age.”

And yet 25 years after the fact, They Might Be Giants have parlayed hard work, a special connection to their fans and a slew of deliriously catchy songs into a thriving career.

Back in 1986, Bar/None Records, saw something in Flansburgh and Linnell that had escaped the attention of the major leagues. “I was impressed with John and John when I first met them,” recalls Bar/None Records co-founder Glenn Morrow. “They had great songs and great ideas about how to present themselves. John Flansburgh would go around nailing copies of their first homemade release to telephone poles.”

They Might Be Giants’ brilliant self-titled debut came out on Bar/None in 1986 and gradually reached the attention of MTV through word of mouth. Soon they were a so-called “buzz band” and destined for bigger things. The subsequent fame that came in 1990 with their major label outing “Flood” and charting single “Birdhouse in Your Soul” proved gratifying but ultimately fleeting. Even more creatively ambitious efforts failed to connect to a mass audience, and soon Flansburgh and Linnell found themselves at the same kind of existential crossroads that unraveled so many of their peers in the aftermath of the alt-rock boom.

But instead of scuttling the band, TMBG happened upon a surprising new outlet for their remarkable ingenuity. In 2002, the band released “No!” a full-length record for children that captured all of the cheerful instrumentation of their prior releases with only a child-appropriate hint of the band’s frequently misanthropic lyrical sentiment. To their astonishment, it outsold their adult records 2 to 1 and captured the attention of Disney Sound, which put them under contract for three additional children’s albums.

The duo had once again identified an unrecognized niche. This one reflected the needs of an audience of young parents desperately seeking substantive entertainment for their offspring. “It was founded on lighthearted fun,” says Linnell. “We weren’t thinking we were going to revolutionize kids’ music or anything. We were doing something in the spirit of everything else we’d done.”

If one great tribute to They Might Be Giants is their devoted fan base, another is the admiration of their peers. Scott McCaughey, sideman to R.E.M. and founder of The Baseball Project and the Young Fresh Fellows, once toured with TMBG and soon found his band’s name mentioned in one of their songs. “I thought we had achieved a modicum of notoriety with the Young Fresh Fellows,” McCaughey recalls. “And suddenly we became, ‘Oh, you’re the band in that They Might Be Giants song.’ ” McCaughey adds, “I was proud of that!”

Harvey Danger frontman and longtime friend and admirer Sean Nelson fell quickly into TMBG’s thrall. “For this band to emerge in the Reagan era, at the height of both punk orthodoxy and music-biz garishness made the anti-rock part of their gesture almost radical and the pop part of their gesture completely ingenious,” Nelson enthuses.

Writers in other genres likewise feel a kinship to TMBG’s music. Author and satirist John Hodgman, best known for his appearances on “The Daily Show,” says, “There is naturally an affinity between my work and the Giants’ as, for many years, and long before I knew them, I have been directly copying them. They are absolutely fearless, and absolutely unconcerned with ‘cool,’ which is of course, the apex of cool.”

Tellingly, TMBG’s latest work ranks among their best. On their most recent release, “Join Us,” Flansburgh and Linnell re-engage with a sound reminiscent of their earliest material. Energetic tracks like “When Will You Die?” and “Judy Is Your Viet Nam” find the band with their caustic edge well intact, rendering music plainly intended for a mature audience.