In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, the Kleisli category of a monad or its Eilenberg-Moore category. This paper elaborates two new unifying ideas: (1) coalgebraic trace semantics is naturally presented in terms of corecursive algebras, and (2) all three approaches arise as instances of the same abstract setting. Our perspective puts the different approaches under a common roof, and allows to derive conditions under which some of them coincide.