  In this thesis we consider the maximal subalgebras of the exceptional Lie
algebras in algebraically closed fields of positive characteristic. This begins
with a quick recap of the article by Herpel and Stewart which considered the
Cartan type maximal subalgebras in the exceptional Lie algebras for good
characteristic, and then the article by Premet considering non-semisimple
maximal subalgebras in good characteristic.
  For $p=5$ we give an example of what appears to be a new maximal subalgebra
in the exceptional Lie algebra of type $E_8$. We show that this maximal
subalgebra is isomorphic to the $p$-closure of the non-restricted Witt algebra
$W(1;2)$.
  After this, we focus completely on characteristics $p=2$ and $p=3$ giving
examples of new non-semisimple maximal subalgebras in the exceptional Lie
algebras. We consider the Weisfeiler filtration associated to these maximal
subalgebras and leave many open questions. There are one or two examples of
simple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.
