Given a group automorphism ϕ: Γ → Γ, one has an action of Γ on itself by ϕ-twisted conjugacy, namely, g.x = gxϕ(g−1). The orbits of this action are called ϕ-conjugacy classes. One says that Γ has the R∞-property if there are infinitely many ϕ-conjugacy classes for every automorphism ϕ of Γ. In this paper we show that any irreducible lattice in a connected semisimple Lie group having finite centre and rank at least 2 has the R∞-property.