  Let $R$ be an order in an algebraic number field. If $R$ is a principal
order, then many explicit results on its arithmetic are available. Among
others, $R$ is half-factorial if and only if the class group of $R$ has at most
two elements. Much less is known for non-principal orders. Using a new
semigroup theoretical approach, we study half-factoriality and further
arithmetical properties for non-principal orders in algebraic number fields.
