A language L ⊆ A∗ is literally idempotent in case that uav ∈ L if and only if uav ∈ L for each u, v ∈ A∗, a ∈ A. Such classes result naturally by taking all literally idempotent languages in a classical (positive) variety or by considering a certain closure operator. We initiate their systematic study. Various classes of such languages can be characterized using syntactic methods. A starting ex...