Always carry your passport or a good colour photocopy with you. Immigration slips are no longer issued or used. The police in Odessa, as in all of Ukraine, are notoriously corrupt and constantly on the look out for tourists to harass with the aim of fining them for breaking some imagined rule or law. Use common sense and caution around rowdy groups and drunks in the city, unless you speak good Russian.
Be very careful in the Arkadia district at night, as it might be not safe in the darker areas. Try to be with someone who knows the clubs and the places and speaks Russian.
