Severe weather, with heavy rainfall and strong winds, has been the cause of recent dramatic land and coastal flooding, and of strong beach and cliff erosion along the British coast. Both the winters of 2012-2013 and 2013-2014 have seen severe environmental disasters in the UK. The prediction of severe rainfall and storms and its use to forecast river flooding and storm surges, as well as coastal erosion, poses a significant challenge. Uncertainties in the prediction of where and how much precipitation will fall, how high storm surges will be and from which direction waves and wind will attack coast lines, lie at the heart of this challenge. This and other environmental challenges are exacerbated by changing climate and need to be addressed urgently. As the latest IPCC reports confirms, sea level rise and storm intensity combined are very likely to cause more coastal erosion of beaches and cliffs, and of estuaries. However, it is also clear that there remains considerable uncertainty. To address the challenges posed by the prediction and mitigation of severe environmental events, many scientific and technical issues need to be tackled. These share common elements: phenomena involving a wide range of spatial and temporal scales; interaction between continuous and discrete entities; need to move from deterministic to probabilistic prediction, and from prediction to control; characterisation and sampling of extreme events; merging of models with observations through filtering; model reduction and parameter estimation. They also share a dual need for improved mathematical models and for improved numerical methods adapted to high-performance computer architectures. Since all these aspects are underpinned by mathematics, it is clear that new mathematical methods can make a major contribution to addressing the challenges posed by severe events. To achieve this, it is crucial that mathematicians with the relevant expertise interact closely with environmental scientists and with end-users of environmental research. At present, the UK suffers from limited interactions of this type. We therefore propose to establish a new Network - Maths Foresees - that will forge strong ties between researchers in the applied mathematics community with researchers in selected strategic areas of the environmental science community and governmental agencies. The activities proposed to reach our objectives include: (i) three general assemblies, (ii) three mathematics-with-industry style workshops, in which the stakeholders put forward challenges, (iii) focussed workshops on mathematical issues, (iv) outreach projects in which the science developed is demonstrated in an accessible and conceptual way to the general public, (v) feasibility projects, and (vi) workshops for user groups to disseminate the network progress to government agencies.