The answer by Geoffrey Thomas is helpful, but a point made in the text deserves amplification. It could reasonably be said that verificationism lives on. But only because the question of "meaningfulness" has been mostly divorced from it:
None of these philosophers believes in the 'criterion of meaningfulness' that the positivists hankered after.
Science needs to be testable, and the ideas behind verificationism are useful for articulating the conditions for testability, and for distinguishing between productive discussion and unproductive speculation. Consider current debates about string theory to see why this is still important.
But the turn to "meaningfulness" was an overreach. Consider a work of fiction. Nothing in a work of fiction is verifiable or falsifiable, because fiction has no truth conditions at all. It doesn't claim to be a true description of anything, and so it can neither be true nor false. But to say that fiction is therefore "meaningless" massively distorts conventional notions of meaning. There's even a strong case to be made that this line of reasoning applies not only to fiction, but also to huge swaths of pure mathematics. The most extreme version of this view is mathematical fictionalism, but even relatively modest views about the nature of mathematical truth make it difficult to support the notion that mathematical statements meet the verifiability criterion of meaning.
It's true that sometimes philosophical terms do have narrow, technical definitions that diverge from standard usage. Verificationism failed precisely because "meaning" was not given such a narrow, technical definition, and if it had been, it would have been clear that "meaning" was not the best word to use.