The design and construction of doubly-curved structures often reveals to be challenging and can result in complex manufacturing and assembly. A recent strategy to tackle this difficulty consists in exploiting the connection between discrete differential geometry and constructive properties to identify curve networks with good fabrication or mechanical properties. Following this approach, a family of surfaces, called Voss surfaces, is here presented. Among other features, they can be built from flat panels, initially flat straight strips, and hinged connections. These properties arise from the existence of a conjugate network of geodesic curves. Two generation methods are presented to shape discrete Voss surfaces: the first allows their exploration through linear spaces; the second provides a unique solution by means of a direct computation. Applications of the generated shapes to architecture are presented. They illustrate the potential of discrete Voss surfaces for the design of doubly curved structures.