We prove the almost sure existence of pure point spectrum for the two-
dimensional Landau Hamiltonian with an unbounded Anderson-like random
potential, provided that the magnetic field is sufficiently large. For
these models, the probability distribution of the coupling constant i
s assumed to be absolutely continuous. The corresponding density g has
support equal to R, and satisfies sup(lambda is an element of R){lamb
da(3+epsilon)g(lambda)} < infinity, for some epsilon > 0. This include
s the case of Gaussian distributions. We show that the almost sure spe
ctrum Sigma is R, provided the magnetic field B not equal 0. We prove
that for each positive integer n, there exists a field strength B-n, s
uch that for all B > B-n, the almost sure spectrum Sigma is pure point
at all energies E less than or equal to (2n + 3)B - O(B-1) except in
intervals of width O(B-1) about each lower Landau level E-m(B) = (2m 1)B, for m < n. We also prove that for any B not equal 0, the integra
ted density of states is Lipschitz continuous away from the Landau ene
rgies E-n(B). This follows from a new Wegner estimate for the finite-a
rea magnetic Hamiltonians with random potentials.