Let F be a non-Archimedean local field. In the representation theory of GL_n(F), one of the basic problems is to characterize its irreducible representations up to isomorphism. There are many invariants (e.g., epsilon factors, L-functions, gamma factors, depth, etc) that we can attach to a representation of GL_n(F). Roughly, the local converse problem is to find the smallest subcollection of twisted local \gamma-factors which classifies the
irreducible admissible representations of GL_n(F) up to isomorphism.
Since the local gamma-factor is the main ingredient of the converse problem, first, we will try to understand the local gamma-factors and their properties.
Then we will discuss the local converse theorems and related problems.
Key words: Local fields, Reductive groups, Generic representation, Whittaker model, L- and \epsilon-factors, local gamma factor, Local converse theorem