Supernormal conduction (SNC) in excitable cardiac tissue refers to an increase of pulse (or action potential) velocity with decreasing distance to the preceding pulse. Here we employ a simple ionic model to study the effect of SNC on the propagation of action potentials (APs) and the phenomenology of alternans in excitable cardiac tissue. We use bifurcation analysis and simulations to study attraction between propagating APs caused by SNC that leads to AP pairs and bunching. It is shown that SNC stabilizes concordant alternans in arbitrarily long paced one-dimensional cables. As a consequence, spiral waves in two-dimensional tissue simulations exhibit straight nodal lines for SNC in contrast to spiraling ones in the case of normal conduction. Propagation of pulse trains in excitable media is usually characterized by a so-called dispersion curve that gives the velocity of a pulse in a periodic train as a function of its wavelength [1]. In cardiac tissue the dispersion properties are often summarized in the conduction velocity (CV) restitution curve that relates the velocity of an action potential (AP) to its diastolic interval (DI), that is, the time between two consecutive APs (see, e.g., Ref.[2]).Since a pulse typically is followed by a refractory zone of decreased excitability, the pulse velocity monotonically increases with increasing wavelength (=normal dispersion). In the context of excitable cardiac tissue, normal dispersion corresponds to normal conduction. In this paper we focus on the alternative situation where the velocity of a pulse is decreasing with increasing wavelength. In generic excitable media this case is known as anomalous dispersion. For excitable cardiac tissue it corresponds to supernormal conduction (SNC) [3]. Normal and supernormal conduction may appear for a different wavelength in the same systems, then the dispersion curve has to be nonmonotonic.The simplest case of a nonmonotonic dispersion relation is a curve with a single maximum separating anomalous dispersion at long wavelength from normal dispersion at short wavelength. Such behavior has been discovered in various experiments in chemical reaction-diffusion systems [4,5]. It has been found to coincide with attractive long-range interaction between pulses that may lead to stable bound states [6] merging or bunching of pulses [7]. While most excitable cardiac tissues and related models exhibit normal conduction, nevertheless examples of SNC are frequent (for a short review see, e.g., Ref.[8]). SNC is found, for example, in the Iyer-Mazhari-Winslow [9] or in the Drouhard-RobergeBeeler-Reuter (DRBR) model [10,11] employed in the study here, and it is often observed under conditions of low extracellular potassium concentration [12]. Recently, it has been shown to potentiate the onset of alternans in cultured cardiac cell strands [13] and also in generic models of spiral wave formation [14].Alternans is an oscillation in the duration of the AP that usually occurs at short stimulation periods [15]. In a paced cable or tissue alternans ...