We examine various issues relevant to localization in the Anderson model. We show there is more to localization than exponentially localized states by presenting an example with such states but where (x(t) )/t is unbounded for any 6 ) 0. We show that the recently discovered instability of localization under rank one perturbations is only a weak instability.(H u) (n) = g u(n + j ) + V~(n)u(n), I jl=& where the potentials V are identically distributed independent random variables with distribution [2] 1 2n g( ""](x) dx, with g~""t he characteristic function of the interval Many of the claimed proofs of localization show that, for almost all cu, an Anderson model Hamiltonian H has a complete set of normalized eigenvectors [3] (p" obeying (n) (where A is fixed, the n 's are some centers of localization, and the C 's are constants depending on~and m.Our first result is an example that shows that mere "exponential localization' of eigenfunctions in the form of Eq. (2) need not have very strong consequences for the dynamics. We can construct a nonrandom potential V in one dimension with the following: (i) H has a complete set of normalized eigenvectors obeying Eq. (2). (ii) Let (x ) (t) denote (e " Bo, x e " Bo); then for any 6 ) 0,The potential U for this example is PACS numbers: 72. 15.Rn Localization in random media is basic to a variety of physical situations. We wish to report here on a number of rigorous mathematical results that shed light on the phenomenon of localization in the Anderson model. Mathematically complete proofs of our results will appear elsewhere [1]. Our goal here is to describe the ideas behind the results. Throughout, we will consider the Anderson model, that is, the Hamiltonian H on 4 (Z") (namely, on the ddimensional cubic lattice) V(n) = 3 cos(27rnn + 0) + AB"o, ]p (n)[~C e'~"-"' eCondition (3) says that the constants C"of(2) are allowed to grow at a rate which is less than exponential in the distance of the n"'sfrom the origin. SULE is closely related to a dynamical condition, which we call semiuniform dynamical localization (SUDL) (n g) ( We have proven that (3) implies (4) with A arbitrarily close to A, and that if H has simple eigenvalues [6], which we consider on the positive half of the lattice (n0 ), with a Dirichlet (or any other) boundary condition at the origin. The 3 in front of the cosine can be replaced by any number larger than 2, and is chosen so that when A = 0 the problem has a positive Lyapunov exponent [5].The n is an irrational, which is specially chosen so that for suitable time scales T"~~, U is so close to periodic that we can show (x (T")) is large compared to T2The local perturbation A6 o pushes the spectrum to be pure point and forces Eq. (2) to hold. While V is very far from random, it illustrates that Eq. (2) is not enough to restrict dynamics. The main failing in (2) is the total freedom given to the constants C . Indeed, when one thinks of "localization, " one usually thinks of the eigenvectors as being confined, at least roughly, within some typical length s...