In this paper, the recent progress on sinuous antennas is detailed, focusing the attention on the antenna geometry, dielectric structure, and miniaturization techniques. In the first part, we introduce the basic principles of the frequency-independent antenna, in particular the self-complementary and log-periodic geometries, as well as the antenna geometries, all characterized in terms of angles. The operating principles, main advantages, system design considerations, limits, and challenges of conventional sinuous antennas are illustrated. Second, we describe some technical solutions aimed to ensure the optimal trade-off between antenna size and radiation behavior. To this aim, some special modification of the antenna geometry based on the meandering as well as on the loading with dielectric structures are presented. Moreover, the cavity backing technique is explained in detail as a method to achieve unidirectional radiation. Third, we present a new class of supershaped sinuous antenna based on a suitable merge of the 2D superformula and the sinuous curve. The effect of the free parameters change on the antenna arm geometry as well as the performance improvement in terms of directivity, beam stability, beam angle, gain, and radiating efficiency are highlighted.