Max-stable processes have been used extensively to model spatial extremes based on block maxima. However, in environmental applications, the spatial dependence strength is often found to weaken as events become more extreme, supporting a property known as asymptotic independence, which max-stable processes cannot capture. In this work, we develop a general class of max-infinitely divisible (max-id) models that can capture asymptotic independence, while keeping well-known max-stable models as special cases. Our new parametric models are based on a general point process characterization, whereby the spatial processes involved have a weaker correlation if their overall magnitude is large. We make inferences by pairwise likelihood, and validate the methodology by simulation and with an application to environmental extremes.