We analyse the probability densities of daily rainfall amounts at a variety of locations on Earth. The observed distributions of the amount of rainfall fit well to a q-exponential distribution with exponent q close to q approximate to 1.3. We discuss possible reasons for the emergence of this power law. In contrast, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q-exponential with q approximate to 1.05 yields the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Frechet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. Looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context.