We demonstrate three-dimensional optical trapping and orientation of individual Au nanorods, Au/Ag core/shell nanorods, and Au bipyramids in solution, using the longitudinal surface-plasmon resonance to enhance optical forces. Laser light that is detuned slightly to the long-wavelength side of the resonance traps individual and multiple particles for up to 20 minutes; by contrast, light detuned to the short-wavelength side repels rods from the laser focus. Under stable-trapping conditions, the trapping time of individual particles depends exponentially on laser power, in agreement with a Kramers escape process. Trapped particles have their long axes aligned with the trapping-laser polarization, as evidenced by a suppression of rotational diffusion about the short axis. When multiple particles are trapped simultaneously, evidence of interparticle interactions is observed, including a nonlinearly increasing two-photon fluorescence intensity, increasing fluorescence fluctuations, and changing fluorescence profiles as the trapped particle number increases.