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In this thesis we consider the maximal subalgebras of the exceptional Liealgebras in algebraically closed fields of positive characteristic. This beginswith a quick recap of the article by Herpel and Stewart which considered theCartan type maximal subalgebras in the exceptional Lie algebras for goodcharacteristic, and then the article by Premet considering non-semisimplemaximal subalgebras in good characteristic.  For $p=5$ we give an example of what appears to be a new maximal subalgebrain the exceptional Lie algebra of type $E_8$. We show that this maximalsubalgebra 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 $p=3$ givingexamples of new non-semisimple maximal subalgebras in the exceptional Liealgebras. We consider the Weisfeiler filtration associated to these maximalsubalgebras and leave many open questions. There are one or two examples ofsimple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.

Maximal subalgebras of the exceptional Lie algebras in low characteristic