We propose a new phase of the successful Mathematics for Real-World Systems (MathSys) Centre for Doctoral Training that will address the call priority area "Mathematical and Computational Modelling". Advanced quantitative skills and applied mathematical modelling are critical to address the contemporary challenges arising from biomedicine and health sectors, modern industry and the digital economy. The UK Commission for Employment and Skills as well as Tech City UK have identified that a skills shortage in this domain is one of the key challenges facing the UK technology sector: there is a severe lack of trained researchers with the technical skills and, importantly, the ability to translate these skills into effective solutions in collaboration with end-users. Our proposal addresses this need with a cross-disciplinary, cohort-based training programme that will equip the next generation of researchers with cutting-edge methodological toolkits and the experience of external end-user engagement to address a broad variety of real-world problems in fields ranging from mathematical biology to the high-tech sector. Our MSc training (and continued PhD development) will deliver a core of mathematical techniques relevant to all applied modelling, but will also focus on two cross-cutting methodological themes which we consider key to complex multi-scale systems prediction: modelling across spatial and temporal scales; and hybrid modelling integrating complex data and mechanistic models. These themes pervade many areas of active research and will shape mathematical and computational modelling for the coming decades. A core element of the CDT will be productive and impactful engagement with end-users throughout the teaching and research phases. This has been a distinguishing feature of the MathSys CDT and is further expanded in our new proposal. MSc Research Study Groups provide an ideal opportunity for MSc students to experience working in a collaborative environment and for our end-users to become actively involved. All PhD projects are expected to be co-supervised by an external partner, bringing knowledge, data and experience to the modelling of real-world problems; students will normally be expected to spend 2-4 weeks (or longer) with these end-users to better understand the case-specific challenges and motivate their research. The potential renewal of the MathSys CDT has provided us with the opportunity to expand our portfolio of external partners focusing on research challenges in four application areas: Quantitative biomedical research, (A2) Mathematical epidemiology, (A3) Socio-technical systems and (A4) Advanced modelling and optimization of industrial processes. We will retain the one-year MSc followed by three-year PhD format that has been successfully refined through staff experience and student feedback over more than a decade of previous Warwick doctoral training centres. However, both the training and research components of the programme will be thoroughly updated to reflect the evolving technical landscape of applied research and the changing priorities of end-users. At the same time, we have retained the flexibility that allows co-creation of activities with our end-users and allows us to respond to changes in the national and international research environments on an ongoing yearly basis. Students will share a dedicated space, with a lecture theatre and common area based in one of the UK's leading mathematical departments. The space is physically connected to the new Mathematical Sciences building, at the interface of Mathematics, Statistics and Computer Science, and provides a unique location for our interdisciplinary activities.