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