Let X={X(t);t∈R+N} be an additive Lévy process in Rd withX(t)=X1(t1)+⋯+XN(tN)∀t∈R+N,where X1,…,XN are independent, classical Lévy processes on Rd with Lévy exponents Ψ1,…,ΨN, respectively. Under mild regularity conditions on the Ψi's, we derive moment estimates that imply joint continuity of the local times in question. These results are then refined to precise estimates for the local and uniform moduli of continuity of local times when all of the Xi's are strictly stable processes with the same index α∈(0,2].