I'm going to give a short, very simplified overview of something that I'm somewhat familiar with, though there are many other open streams in research that I don't have the experience to comment on.
Let $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ denote the absolute Galois group, i.e. the group of all field automorphisms $\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}$ fixing the rationals. Equivalently, $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the inverse limit of Galois groups $\text{Gal}(L/\mathbb{Q})$ of finite Galois extensions $L/\mathbb{Q}$, so in a certain sense it is made up of all finite Galois groups over $\mathbb Q$.
Perhaps the most well-known open problem in Galois theory is
What is the structure of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$?
An approach to this problem is through the famous Langlands program. A different approach was outlined by Grothendieck in his, also relatively well known, Esquisse d'un programme.
There Grothendieck notes that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has a faithful action on the collection of graphs embedded on compact surfaces, which he calls dessins d'enfants (children's drawings) due to their apparent simplicity. If one can understand this action, then in principle one can represent the elements of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ as permutations of dessins d'enfants. Thus one of the main open problems of the theory of dessins d'enfants is to
Classify enough invariants of dessins d'enfants so that any two orbits of the action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ can be distinguished.
Shortly after Grothendieck's Esquisse went into circulation, Drinfeld proved that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ injects into the so-called Grothendieck-Teichmuller group, which has an explicit description in terms of generators and relations. Hence one other open problem is
Is $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ isomorphic to the Grothendieck-Teichmuller group?
These are very difficult problems which raise further questions still unresolved, for example how can one compute a dessin d'enfant efficiently. Furthermore, theoretical physicists are also interested in Galois theory: Drinfeld's introduction of the Grothendieck-Teichmuller group was motivated by mathematical physics, and dessin d'enfants have already appeared in physics under a different name, dimer models.
