Let $E$ be an arbitrary directed graph and let $L$ be the Leavitt path algebra of the graph $E$ over a field $K$. The necessary and sufficient conditions are given to assure the existence of a maximal ideal in $L$ and also the necessary and sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shown that if a maximal ideal $M$ of $L$ is nongraded, then the largest graded ideal in $M$, namely $gr(M)$, is also maximal among the graded ideals of $L$. Moreover, if $L$ has a unique maximal ideal $M$, then $M$ must be a graded ideal. The necessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed.