At very high temperature electrons leave the atom to which they are

attached and a gas of charged particles called a plasma is obtained.

Plasmas, because of their interaction with an electromagnetic field,

have a global and rich behaviour, much more complex than for neutral

gases. This makes their study complex and fascinating. Plasma

physics and its lighthouse application, controlled thermonuclear

fusion, thus offers new challenges to mathematicians. Two main

models are generally used to describe the dynamics of a plasma :

on the one hand the so-called kinetic models which describe the

evolution of the phase space density of the different particle species,

and on the other hand the fluid models which describe the evolution of

macroscopic quantities, like density in physical space, mean velocity

and temperature. This two types of models need to be coupled with

Maxwell's équations in order to describe the non linear interactions

with the electromagnetic field generated by the charged particles of

the plasma. This book results from notes of the lectures that were

given at CIRM in Luminy during the summer school of CEMRACS

2010. It détails for topics in the mathematical and 'numerical study

of plasmas : asymptotic preserving (AP) numerical schemes that

enable to deal consistently with two space or time scales, gyrokinetic

simulations of magnetic fusion plasmas, MHD simulations of magnetic

fusion plasmas and the mathematical study of Landau damping.