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Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: https://arxiv.org/abs/1812.10454.

This was probably considered the biggest open problem in the combinatorics of simplicial complexes. See Gil Kalai's blog post: https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/.

Hinged dissections exist.
(See 3-piece dissection of square to equilateral triangle? for an animation of Dudeney's famous equilateral-triangle-to-square hinged dissection.)

Abbott, Timothy G., Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers. "Hinged dissections exist." Discrete & Computational Geometry 47, no. 1 (2012): 150-186.
Springer link.

"Abstract. We prove that any finite collection of polygons of equal area has a common hinged dissection. That
is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be
folded in the plane continuously without self-intersection to form any polygon in the collection. This
result settles the open problem about the existence of hinged dissections between pairs of polygons that
goes back implicitly to 1864 and has been studied extensively in the past ten years."

The homological conjectures in commutative algebra using perfectoid methods. A survey on many recent developments written by André can be found here.

Konstantin Tikhomirov recently proved that the probability that a random $ntimes n$ Bernoulli matrix $M_n$ with independent $pm 1$ entries, and $$mathbb{P}[M_{ij}=1]=p,quad 1leq i,jleq n,$$ is singular is
$$
mathbb{P}[M_n~mathrm{is~singular}]=(1-p+o_n(1))^n
$$
for any fixed $pin (0,1/2].$

This problem was considered by Komlos, Kahn-Komlos-Szemeredi, Bourgain, Tao-Vu etc., so I am unsure if it qualifies in terms of being not-so-famous.

Nevertheless it was exciting reading about it in Gil Kalai's blog here .

The Audin conjecture in symplectic topology, posed in 1988 by Audin in her famous paper on Lagrangian immersions, asserts that all Lagrangian tori in the standard symplectic vector space have minimal Maslov number 2. This was recently proven by Cieliebak and Mohnke:

https://arxiv.org/abs/1411.1870

That paper nicely summarises the history of the conjecture:

"This question was answered earlier for n = 2 by Viterbo [57] and Polterovich [54], in the monotone case for n ≤ 24 by Oh [52], and in the monotone case for general n by Buhovsky [12] and by Fukaya, Oh, Ohta and Ono [28, Theorem 6.4.35], see also Damian [22]. A different approach has been outlined by Fukaya [27]. The scheme to prove Audin’s conjecture using punctured holomorphic curves was suggested by Y. Eliashberg around 2001. The reason it took over 10 years to complete this paper are transversality problems in the non-monotone case."

The strong no loop conjecture for quiver algebras $A$ states that a simple module $S$ with $Ext_A^1(S,S) neq 0$ has infinite projective dimension. It was proven here https://www.sciencedirect.com/science/article/pii/S0001870811002714 .
The more general conjecture for Artin algebras is still open.

(The result can be used to check for finite global dimension of endomorphism algebras, see for example Does this algebra have finite global dimension ? (Human vs computer).)

Ladner's theorem states that there exist $mathsf{NP}$-intermediate problems when $mathsf{P}neqmathsf{NP}$. However, the problem constructed in Ladner's proof is rather 'unnatural'. The question arises of whether any 'natural' examples of problems can be $mathsf{NP}$-intermediate.

The Dichotomy Conjecture of Feder and Vardi (first stated here) states that, under the assumption that $mathsf{P}neqmathsf{NP}$, the computational problems known as constraint satisfaction problems (CSPs for short) are either $mathsf{NP}$-complete or belong to $mathsf{P}$.

The consensus in the community (last I knew) is that Dmitriy Zhuk (https://arxiv.org/abs/1704.01914) and Andrei Bulatov (https://arxiv.org/abs/1703.03021) have independently proven the conjecture to be true. Their proofs cap a decades long approach of applying universal algebra to the question.

The Hall-Paige conjecture, first posed in 1955 by Marshall Hall and L. J. Paige, is the following:

A finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are not cyclic.

Note that a complete mapping is a bijection $phi : G to G$ such that the function given by $psi(g) = g phi(g)$ is also a bijection. The above statement was shown to be necessary by Hall and Paige, but its sufficiency remained open until very recently; in 2009, it was shown to be sufficient to only check the cases when $G$ is a finite simple group, and the same year all finite simple groups except for $J_4$ were shown to satisfy the conjecture. John Bray then dealt with this final case in unpublished work, and Peter Cameron was able to convince him (see this) to publish these noteworthy calculations many years later; the final proof of the Hall-Paige conjecture, together with some consequences of it regarding synchronicity in groups, was written up in 2018 and can be found as a preprint on the arXiv.

Graph theory / Discrete dynamics: In 2007, A. Trahtman proved the Road Coloring Conjecture, which had been posited 37 years earlier by R. Adler and B. Weiss.