EUV and X-ray observations of late-type stars. HodgkinSimon T. 2015 In this thesis I describe a number of projects arising from the ROSAT mission, inspired by a desire to understand better the activity of late-type stars from studies of their coronal EUV and X-ray emission. A brief introduction summarises some of the most important work on cool star coronas. The second chapter describes the mechanisms by which X-rays are produced in coronal plasmas. I also discuss the ROSAT mission, its instrumentation and applicability for observing cool stars. In Chapter 3 I describe the discovery of one of the brightest sources in the EUV sky, the hot white dwarf companion to HD 33959C. I discuss the importance of such binaries for the determination of more accurate measurements of mass, radius and distance than is possible for isolated white dwarfs. Chapter 4 is a WFC survey of all known late-type stars within 25 parsecs of the Sun. I construct and discuss the first ever EUV luminosity functions for such a sample and show that stars in interacting binaries are more active than single stars. In Chapters 5 and 6 I present a deep PSPC survey of the Hyades, comprising 11 overlapping pointings. In Chapter 5 I derive the Hyades dK and dM X-ray luminosity functions down to Lx ~ 5 x 1027 erg s-1. In Chapter 6 I investigate the X-ray spectra of the more luminous Hyads, both non-parametrically using hardness ratios, and parametrically using simple one-temperature and two-temperature fits to the data. Flares were observed in VB 50 and VA 334; both stars show increases in temperature and emission measure during the flaring episodes. In Chapter 7 I summarise my conclusions and discuss projects which arise from the work presented in this thesis. Finally I describe some of the impacts that we may expect to see from three future missions, JET-X, AXAF and XMM.