Let $X$ be a smooth projective geometrically irreducible curve over $\mathbb F_{q}$ a finite field, $F$ its global field and $\mathbf G/F$ a split connected reductive group and $\widehat{\mathbf G}$ its Langlands dual. Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathbf G(\mathbb A_{F})$. Then for all $\ell\nmid q$ there exists a $G_{F}$-representation $\sigma_{\pi,\ell}$ with values in $\widehat{\mathbf G}(\bar{\mathbb Q}_\ell)$ attached to $\pi$ in the usual sense. This result is essentially optimal, so that the answer to the question in the title seems to be "all of them".

This is a result of Laurent Lafforgue (Invent. Math. 147) when $\mathbf G=GL_{n}$ and of Vincent Lafforgue (preprint available from his webpage) in the other cases.

That said, these results are far beyond my own expertise, so I hope real experts will chime in.