We show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≥ κ and $\lambda^{ then there is a nontrivial κ-complete κ-normal fine ideal on P κ (λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ + and μ is a regular cardinal then (1) $\mu^{ if (...) and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if $\mu^{ then there is a nontrivial κ-complete weakly κ-normal fine ideal on P κ (λ). (shrink)