We study the influence of a slow classical noise on Landau-Zener transitions.  We demonstrate that the slow colored noise characterized by correlation time much greater compared to Landau-Zener time can produce a transition itself, or can change substantially the Landau-Zener transition probabilities.  Exact formulas for the time-evolution operator and finite time probabilities are derived. Connections between the slow noise induced Landau-Zener transitions and recent experiments on spin blockade in nano-devices are discussed.