We consider finite-dimensional real vector spaces X ordered by a closed cone X+ with non-empty interior and study eventual nonnegativity of matrix semigroups (etA)t≥0 with respect to this cone.Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that etA maps X+ into X+ for all sufficiently large times t; (ii) individual eventual nonnegativity means that for each x∈X+ the vector etAx is in X+ for all t larger than an x-dependent time t0; and (iii) weak eventual nonnegativity means that for each x∈X+ and each functional x′ in the dual cone X+′ the value 〈x′,etAx〉 is in [0,∞) for all t larger than an x- and x′-dependent time t0. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different.Our second contribution is to show that typical Perron–Frobenius like properties remain valid under the weakest of the above notions.Third, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that the uniform, individual and weak versions of this property are – in contrast to the nonnegative case – equivalent, and that they can be characterized by spectral properties.