The continuum hypothesis. Of course it's extremely famous, but everyone thinks it's resolved. I was astonished to find out that some serious set theorists apparently consider it (I mean in the present, decades past Cohen's proof) to be an important open problem that people should be working on ...

Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable? Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable? Is there a universal Diophantine equation of power $3$? Is there a universal Diophantine equation containing less than $9$ variables? I...

Some pages: Open Problem Garden The Open Problems Project

The Feit-Thompson conjecture is not too famous and it is still open: there are no prime numbers $p\neq q$ such that $$\frac{p^q-1}{p-1}\text{ divides }\frac{q^p-1}{q-1}.$$

Farideh Firoozbakht conjecture states that the sequence $P_n^{1/n}$ is strictly decreasing, where $P_n$ is the $n-$th prime number. This conjecture can also give simple proofs to many theorems related to Prime numbers. It is not proved yet.

I like Moser's worm problem: https://en.wikipedia.org/wiki/Moser%27s_worm_problem I'm not sure how famous it is, but I believe less than it deserves. In short: what is the area of the smallest planar region such that every curve of unit length can be placed into it?

Does the decimal expansion of $2^n$ contain the digit $7$ for all sufficiently large $n$?

If $2^x$ and $3^x$ are integers for some positive real number $x$, does this imply that $x\in\mathbb{N}$?

It is not possible to find three corners of a square that form an equilateral triangle. (Left as an exercise for the reader.) It is, however, possible to find four corners of a cube that form a regular tetrahedron:                                            We could ask: in which dimensions $n>1$...

I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra can be seen as a $C^*$-algebra structure, but I am unsure whether this structure will be compatible...