Elastic and inelastic scattering, one-nucleon transfers [mostly ${(}^{8}$Li, $^{7}\mathrm{Li}$)] and the ${(}^{8}$Li,\ensuremath{\alpha}) fusion-evaporation reaction have been studied on a variety of targets (A=1--58) at E${(}^{8}$Li)\ensuremath{\approxeq}13--20 MeV using the University of Michigan--University of Notre Dame radioactive nuclear beam (RNB) facility. The latter, which produces an energy-resolved (\ensuremath{\sim}500 keV FWHM), focused beam with intensity \ensuremath{\gtrsim}${10}^{6}$ $^{8}\mathrm{Li}$/s, permitted measurements to selected nuclear levels with cross sections down to approximately 1 mb/sr. Based on the systematics of the measured $^{8}\mathrm{Li}$ elastic scattering an appropriate $^{8}\mathrm{Li}$ optical potential was deduced. Large inelastic scattering cross sections (and B[E2\ensuremath{\uparrow}]) are observed for $^{8}\ensuremath{\rightarrow}^{8}$${\mathrm{Li}}_{0.98}^{\mathrm{*}}$. Large cross sections are also observed for the ${(}^{8}$Li${,}^{7}$Li) transfer reaction corresponding to quasielastic (viz., Q\ensuremath{\approxeq}0) neutron transfer to selected high spin excited levels in the residual nucleus. The ${(}^{8}$Li,\ensuremath{\alpha}) fusion-evaporation reaction also appears to be significant and can produce very-high-energy \ensuremath{\alpha} particles, viz., ${\mathit{E}}_{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\gg}${\mathit{E}}_{\mathrm{beam}}$. Although observed using $^{8}\mathrm{Li}$, these features appear to be important for many RNB-induced reactions and they must be considered, for example, in nucleosyntheis calculations involving $^{8}\mathrm{Li}$ and other short-lived radioactive nuclei.