In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the following

Conjecture: (Fontaine-Mazur) Suppose that $$R\colon G_{\mathbf{Q}}\rightarrow \mathrm{GL}(V),$$ is an irreducible $\ell$-adic representation which is unramified at all but finitely many primes and with the $R|_{G_{\mathbf{Q}_\ell}}$ de Rham. Then there is a smooth projective variety $X/\mathbf{Q}$ and integers $i\ge 0$ and $j$ such that $V$ is a subquotient of $H^i(X(\mathbf{C}),\overline{\mathbf{Q}}_\ell(j))$. In particular $R$ is pure of some weight $w\in \mathbf{Z}$.

(here $G_K$ means absolute galois group of $K$ etc.) The notion of a de Rham representation is rather long - it may be found e.g. in the lecture notes of O. Brinon and B. Conrad here, and see loc. cit. for explanations of the other conditions. The references for the conjecture are Fontaine (J.M. Fontaine talk at ''Mathematische Arbeitstagung 1988'', Max-Planck-Institut für mathematik preprint no. 30 of 1988) and Fontaine-Mazur

Question: What is the current status of this conjecture? What results are known in its direction?

There seems to be a lot of research papers published in the last number of years on this topic. I would be grateful if anyone would be able to present some kind of rough panorama of results related to the conjecture.