In preparation for the Astro 2020 Decadal Survey NASA has commissioned the study four flagship missions spanning to a wide range of observable wavelengths: The Origins Space Telescope (OST, formerly the Far-Infrared Surveyor), Lynx (formerly the X-ray Surveyor), the Large UV/Optical/Infrared Surveyor (LUVOIR) and the Habitable Exoplanet Imager (HabEx). One of the key scientific objectives of the latter two is the detection and characterization of the earth-like planets around nearby stars using the direct imaging technique (along with a broad range of investigations regarding the architecture of and atmospheric composition exoplanetary systems using this technique). As a consequence dedicated exoplanet instruments are being studied for these mission concepts. This paper discusses the design of the coronagraph instrument for the architecture "A" (15 meters aperture) of LUVOIR. The material presented in this paper is aimed at providing an overview of the LUVOIR coronagraph instrument. It is the result of four months of discussions with various community stakeholders (scientists and technologists) regarding the instrument's basic parameters followed by meticulous design work by the the GSFC Instrument Design Laboratory team. In the first section we review the main science drivers, presents the overall parameters of the instrument (general architecture and backend instrument) and delve into the details of the currently envisioned coronagraph masks along with a description of the wavefront control architecture. Throughout the manuscript we describe the trades we made during the design process. Because the vocation of this study is to provide a baseline design for the most ambitious earth-like finding instrument that could be possibly launched into the 2030's, we have designed an complex system privileged that meets the ambitious science goals out team was chartered by the LUVOIR STDT exoplanet Working Group. However in an effort to minimize technological risk we tried to maximize the number of technologies that will be matured by the WFIRST coronagraph instruments.