In this thesis we explore the gradings by groups on the simple Cartan type Lie algebras and Melikyan algebras over algebraically closed fields of positive characteristic p > 2 (p = 5 for the Melikyan algebras).;We also give examples of gradings by the cyclic group of order p which do not follow the pattern of the general description of gradings by groups without p-torsion as well as describe gradings by arbitrary groups on the restricted Witt algebra W(1; 1).;We approach the gradings by abelian groups without p-torsion on a simple Lie algebra L by looking at the dual group action. This action defines an abelian semisimple algebraic subgroup (quasi-torus) of the automorphism group of L. A result of Platonov says that any quasi-torus of an algebraic group is contained in the normalizer of a maximal torus. We show that if L is a simple graded Cartan or Melikyan type Lie algebra, then any quasi-torus of the automorphism group of L is contained in a maximal torus. Thus all gradings by groups without p-torsion are, up to isomorphism, coarsenings of the eigenspace decomposition of a maximal torus in the automorphism group.
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