  We study fragments of first-order logic and of least fixed point logic that
allow only unary negation: negation of formulas with at most one free variable.
These logics generalize many interesting known formalisms, including modal
logic and the $\mu$-calculus, as well as conjunctive queries and monadic
Datalog. We show that satisfiability and finite satisfiability are decidable
for both fragments, and we pinpoint the complexity of satisfiability, finite
satisfiability, and model checking. We also show that the unary negation
fragment of first-order logic is model-theoretically very well behaved. In
particular, it enjoys Craig Interpolation and the Projective Beth Property.
