We want to present a critical discussion about the possible existence of atomically thin boron structures, such as boron nanotubes and fullerenes. Despite recent successes in the synthesis of boron nanotubes (BNTs) many questions about their atomic structure and other basic properties do remain. Systematic theoretical studies on boron fullerenes suggested that such fullerenes are unlikely to exist. Recently, we unveiled the dynamical stability of BNTs by means of first-principles molecular dynamics simulations. We found that free-standing, single-wall BNTs with diameters larger than 0.6 nm are thermally stable at the experimentally reported synthesis temperature of 870C and higher. The walls of thermally stable BNTs are found to have a variety of different mixed triangular- hexagonal morphologies. Our results indicate that mixed triangular-hexagonal morphologies are structural paradigm for atomically thin boron. Furthermore, our recent experimental results on the room temperature synthesis of boron nanowires and boron oxide nanotubes will be presented.