A long term goal of Artificial Intelligence (AI) has been to create machines that can understand spoken and written human language. This capability would enable, for example, spoken language interaction between people and computers, translation between all human languages and tools to analyse and answer questions about vast archives of text and speech. Spectacular advances in computer hardware and software over the last two decades mean this vision is no longer science fiction but is turning into reality. Speech and Language Technologies (SLTs) are now established as core scientific/engineering disciplines within AI and have grown into a world-wide multi-billion dollar industry, with SLT global revenues predicted to rise from $33bn in 2015 to $127bn by 2024. The UK has long played a leading role in SLT and the government has recently identified AI, including SLT, as of national importance. Many international corporations such as Google, Apple, Amazon and Microsoft now have research labs in the UK, in part to leverage local SLT expertise, and a new and extensive eco-system of SLT SMEs has sprung up. There is huge demand for scientists with advanced training in SLT from these organisations, most of whom hire only at PhD level, evident in the support for this CDT by more than 30 partners. The result is fierce, international competition to attract talent and supply is falling far short of demand. It is critically important, therefore, to improve the UK's capacity to address this industrial need for high quality, high value postdoctoral SLT talent, to enhance the UK's position as a leader in the field and, in turn, attract investment in AI-related technologies and support UK economic growth. To address the shortfall in PhD-trained scientists we propose a CDT in "Speech and Language Technologies and Their Applications". Our vision is to create a CDT that will be a world-leading centre for training SLT scientists and engineers, giving students the best possible advanced training in the theory and application of computational speech and language processing, in a setting that fosters interdisciplinary approaches, innovation and engagement with real world users and awareness of the social and ethical consequences of our work. A cohort-based approach is necessary in SLT because: (1) the software infrastructure, tools and methods for SLT are highly complex and creating them is nearly always a collaborative endeavour -- a cohort offers an ideal setting to gain experience of such collaborative working (2) PhD topics tend to be narrow and focused on specifics and do not include the broad overview needed in students' later careers -- through cohort training we can expose students to a range of different SLT topics (3) peer learning within and across cohorts is a highly effective way to hand over tools and to teach methodology (4) a multi-year cohort programme allows significant and sustained progression in larger (i.e. multi-student) SLT projects, resulting in better research outcomes and more impact in partnering companies (5) cohort teaching is very attractive to students (6) an extended cohort-based training programme with strong group work and peer tutoring elements allows students with non-standard backgrounds be admitted, helping to promote diversity in SLT. To realise our vision we propose to build on Sheffield's unique strengths in SLT, which include (1) a large team of SLT academics with an outstanding, 30-year research track record in publication, research grant capture and PhD supervision, covering all the core areas of SLT (2) a large group of industrial partners who actively want to participate in the CDT (3) a track record of impact arising from our research, through creating new enterprises or enhancing the activities of existing organisations (4) an excellent research environment in terms of computing and data resources, study and work facilities, and commitment to and respect for diversity and equality.