This work gives a full description of a method for analyzing the admissible complex representations of the general linear group *G* = *Gl(N,F)* of a non-Archimedean local field *F* in terms of the structure of these representations when they are restricted to certain compact open subgroups of *G*. The authors define a family of representations of these compact open subgroups, which they call *simple types.* The first example of a simple type, the "trivial type," is the trivial character of an Iwahori subgroup of *G*. The irreducible representations of *G* containing the trivial simple type are classified by the simple modules over a classical affine Hecke algebra. Via an isomorphism of Hecke algebras, this classification is transferred to the irreducible representations of *G* containing a given simple type. This leads to a complete classification of the irreduc-ible smooth representations of *G*, including an explicit description of the supercuspidal representations as induced representations. A special feature of this work is its virtually complete reliance on algebraic methods of a ring-theoretic kind. A full and accessible account of these methods is given here.