We prove that the Banach space,(n=1 lnp)lq, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1p and 1q. Furthermore, the Banach spaces(n=1 lnp)lq, with 1p, and(n=1 lnp)lq, with 1p, do not have a greedy basis. We prove as well that the space(n=1 lnp)lq has a 1-greedy basis if and only if 1p=q. 2010 Springer Science+Business Media, LLC.