The paper introduces the concept of distributed lag non-linear models (DLNMs), a statistical framework designed to model the delayed and non-linear effects of environmental stressors on health outcomes. DLNMs extend traditional distributed lag models (DLMs) by incorporating non-linear relationships, allowing for a more flexible representation of the exposure-response curve. The authors define a 'cross-basis' as a bi-dimensional space of functions that captures both the predictor space and the lag dimension, enabling the modeling of complex temporal dependencies. The methodology is implemented in the R package `dlm`. The paper provides a detailed theoretical framework, including the algebraic representation of DLNMs and their application to a real-world dataset from New York City, where the effect of temperature on mortality is analyzed. The results highlight the potential of DLNMs to capture intricate relationships between environmental factors and health outcomes, particularly in the context of delayed and non-linear effects.The paper introduces the concept of distributed lag non-linear models (DLNMs), a statistical framework designed to model the delayed and non-linear effects of environmental stressors on health outcomes. DLNMs extend traditional distributed lag models (DLMs) by incorporating non-linear relationships, allowing for a more flexible representation of the exposure-response curve. The authors define a 'cross-basis' as a bi-dimensional space of functions that captures both the predictor space and the lag dimension, enabling the modeling of complex temporal dependencies. The methodology is implemented in the R package `dlm`. The paper provides a detailed theoretical framework, including the algebraic representation of DLNMs and their application to a real-world dataset from New York City, where the effect of temperature on mortality is analyzed. The results highlight the potential of DLNMs to capture intricate relationships between environmental factors and health outcomes, particularly in the context of delayed and non-linear effects.