Carbon nanotube tapers are a set of nanostructures comprised of straight tubular sections with decreasing diameters, joined to each other via conical funnels and terminated with a hemispherical cap. The funnels are formed with the help of topological defects, which minimally include at least one pentagon-heptagon pair. The structural, electronic, and transport properties of tapers are analyzed using realistic tight-binding models. Specifically, it is shown that straight nanotube tapers are monochiral objects. Among a variety of possible taper structures, kinetics of the growth process suggests that the most prevalent tapers will have either zigzag or armchair structures. Their scanning tunneling microscopy STM images have been simulated for identification purposes. The STM images of tapers are dominated by a protruding pentagon inherent in the taper structure, which unfortunately does not allow for an easy identification of the chirality of the underlying nanotubes. Turning to transport properties, it is shown that zigzag-based tapers will likely be poor conductors, because of gaps induced by the semiconducting segments. Armchair-based tapers, on the other hand, are characterized by a finite conductance at low bias voltages and make attractive prototypes for nanoscale probes and devices.