We present a new class of boron sheets, composed of triangular and hexagonal motifs, that are more stable than structures considered to date and thus are more likely to be the precusors of boron nanotubes. We describe a simple and clear picture of electronic bonding in boron sheets and highlight the importance of three-center bonding and its competition with two-center bonding, which can also explain the stability of recently discovered boron fullerenes. Our findings call for reconsideration of the literature on boron sheets, nanotubes and clusters.PACS numbers: 73.63.Fg All boron nanotubes (BNT), regardless of diameter or chirality, are predicted to be metallic and have large densities of states (DOS) at their Fermi energies (E F ) [1]. In contrast, carbon nanotubes (CNT) can be semiconductors or metals with small DOS at their E F . Metallic CNT are used widely to study one-dimensional (1D) electronics [2,3] and are superconducting at low temperatures [4,5]. Due to the larger DOS, BNT should be better metallic systems for 1D electronics and may have higher superconducting temperatures than CNT.Recent experiments have fabricated boron nanotubular structures both as small clusters [6] and long, 1D geometries [7]. Understanding the properties of BNT is crucial for realizing their applications. For CNT, it has been fruitful to study two-dimensional (2D) graphene: e.g., many properties of CNT are derived directly from graphene [8,9]. For boron, however, no 2D planar structure exists in its crystals which are built from B 12 icosahedra [10]. Researchers have proposed several 2D boron sheets (BS). The hexagonal graphitic BS was found to be unstable [11,18]. Based on extensive theoretical studies of boron clusters [11,12,13,14,15], an Aufbau principle was proposed whereby the most stable structures should be composed of buckled triangular motifs [12] . Experiments on small clusters of 10-15 atoms support this view [16]. A recent study of many possible sheet structures found, again, the buckled triangular arrangement to be most favorable [17]. Hence, 2D triangular BS have been studied and used to construct BNT [18,19,20].In this Letter, we present a class of boron sheets that are more stable than the currently accepted ones. We describe their structures, energetics, electronic states, and provide a clear picture of the nature of their bonding that clarifies their stability. We also show that clusters with these structures are competitive with or more favorable than those considered to date. Our findings have important consequences for understanding and interpreting the properties of these systems. For example, the unusual stability of B 80 fullerenes [21] can be explained by our bonding picture. Hence, in our view, it is necessary to reconsider previous work in this general field.We use Density Functional Theory [22,23] within the ab initio supercell planewave pseudopotential total energy approach [24]. Calculations are done by PARATEC [25]. We use both the local density approximation (LDA) [23,26] and the generalize...