Here are two examples from operator theory/operator algebras. Both were open problems for more than forty years. The first example is remarkable because it is was quite well known, it is so simple to state and was elusive for so much time. The second example is notable because of its importance and because it was open since Arveson's seminal "subalgebras" paper.
In 2015 Kreg Knese proved that von Neumann's inequality holds for triples of $3 \times 3$ contractions (this breakthrough followed important work of Lukasz Kosinski). The introduction to Knese's paper explains it all well (I blogged about it, in case you want a version with some more superlatives).
In 2013 Davidson and Kennedy proved the existence of "sufficiently many boundary representations" in every operator system, which was an open problem raised by Arveson in 1969. Here is their paper on arxiv (here is a blog post I wrote on this, geared towards non-specialists). Davidson and Kennedy's solution came five years after Arveson himself settled the problem for separable operator systems (following important work of Dritschel and McCullough), which was also an exciting development.