Title: Monotonicity as a requirement and as a regularizer: efficient methods and applications. Abstract: We study the setting where risk minimization is performed over general classes of models and consider two cases where monotonicity is treated as either a requirement to be satisfied everywhere or a useful property. We specifically consider cases where point-wise gradient penalties are used alongside the empirical risk during training. In our first contribution, we show that different choices of penalties define the regions of the input space where the property is observed. As such, previous methods result in models that are monotonic only in a small volume of the input space. We thus propose an approach that uses mixtures of training instances and random points to populate the space and enforce the penalty in a much larger region. As a second contribution, we introduce the notion of monotonicity as a regularization bias for convolutional models. In this case, we consider applications, such as image classification and generative modeling, where monotonicity is not a hard constraint but can help improve some aspects of the model. Namely, we show that using group monotonicity can be beneficial in several applications such as: (1) defining strategies to detect anomalous data, (2) allowing for controllable data generation, and (3) generating explanations for predictions. Our proposed approaches do not introduce relevant computational overhead while leading to efficient procedures that provide extra benefits over baseline models.