Natural critical phenomena are characterized by laminar periods separated by events where bursts of activity take place, and by the interrelated self-similarity of space-time scales and of the event sizes. One example are earthquakes: for this case a new approach to quantify correlations between events reveals new phenomenology. By linking correlated earthquakes one creates a scale-free network of events, which can have applications in hazard assessment. Solar flares are another example of critical phenomenon, where event sizes and time scales are part of a single self-similar scenario: rescaling time by the rate of events with intensity greater than an intensity threshold, the waiting time distributions conform to scaling functions that are independent of the threshold. The concept of self-organized criticality (SOC) is suitable to describe critical phenomena, but we highlight problems with most of the classical models of SOC (usually called sandpiles) to fully capture the space-time complexity of real systems. In order to fix this shortcoming, we put forward a strategy giving good results when applied to the simplest sandpile models.