Let p be a prime number and K be a number field. For a given mod p Galois representation \rho:{\rm Gal}(\overline{K}/K)\longrightarrow GL_n(\overline{\mathbb{F}}_p), we may expect the existence of an automorphic representation of G(\mathbb{A}) for some reductive group G which give rise to \rho via conjectural global Langlands correspondence and its reduction. This is called as
Serre's (mod p automorphy) conjecture for (K,\rho,G).
Many people have tried to reformulate this vague statement to specify
possibly corresponding automorphic representations from data which (K,\rho,G) inherits.
In this talk, we give an example-based survey around this topic including speaker's works for Serre's conjecture for (\Q,\rho,GSp_4) where \rho takes the values in GSp_4.