An isocrystal is a type of linear algebraic datum that describes  various objects arising in arithmetic algebraic geometry. On the one hand, they have a relatively concrete description as vector spaces  
over a field together with some special endomorphisms. This makes them  very amenable to computation. On the other, they are linked to very  important structures in Arithmetic Geometry such as p-divisible groups  and the crystalline cohomology of algebraic varieties. This dual  nature makes them indispensable tools for tackling many problems. In  this talk, we will define the category of isocrystals over a field of  characeristic p, explain its basic properties, and give some  
applications.