Standards play an important role in providing a common set of specifications and allowing inter-operability between devices and systems. Until recently, no standard for High Dynamic Range (HDR) image coding had been adopted by the market, and HDR imaging relies on proprietary and vendor specific formats which are unsuitable for storage or exchange of such images. To resolve this situation, the JPEG Committee is developing a new coding standard called JPEG~XT that is backwards compatible to the popular JPEG compression, allowing it to be implemented using standard 8-bit JPEG coding hardware or software. In this paper, we present design principles and technical details of JPEG~XT. It is based on a two-layers design, a base layer containing a Low Dynamic Range (LDR) image accessible to legacy implementations, and an extension layer providing the full dynamic range. The paper introduces three of currently defined profiles in JPEG~XT, each constraining the common decoder architecture to a subset of allowable configurations. We assess the coding efficiency of each profile extensively through subjective assessments, using 24 naive subjects to evaluate 20 images, and objective evaluations, using 106 images with five different tone-mapping operators and at 100 different bit rates. The objective results (based on benchmarking with subjective scores) demonstrate that JPEG~XT can encode HDR images at bit rates varying from 1.1 to 1.9 bit/pixel for estimated mean opinion score (MOS) values above 4.5 out of 5, which is considered as fully transparent in many applications. This corresponds to 23-times bitstream reduction compared to lossless OpenEXR PIZ compression.