Branching ratios of nonleptonic exclusive two-body decays of $B$ mesons without final-state charm particles are studied in detail. The technique used for evaluating the nonleptonic decay amplitudes is elucidated. The dominant charmless decay channels are ${B}_{u}^{+}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\eta}}^{\ensuremath{'}}, {\ensuremath{\pi}}^{+}{\ensuremath{\eta}}^{\ensuremath{'}}, {\ensuremath{\rho}}^{+}{\ensuremath{\pi}}^{0}, {\ensuremath{\rho}}^{+}\ensuremath{\eta}, {\ensuremath{\rho}}^{+}{\ensuremath{\eta}}^{\ensuremath{'}}, {\ensuremath{\rho}}^{+}\ensuremath{\omega}$, ${B}_{d}^{0}\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\eta}}^{\ensuremath{'}}, \ensuremath{\eta}{\ensuremath{\eta}}^{\ensuremath{'}}, {\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}}, {K}^{*+}{\ensuremath{\pi}}^{\ensuremath{-}}, {\ensuremath{\rho}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}, {K}^{*+}{\ensuremath{\rho}}^{\ensuremath{-}}, {\ensuremath{\rho}}^{+}{\ensuremath{\rho}}^{\ensuremath{-}}$, which should be interesting for experimental observation. The implications of the quark-diagram scheme are also discussed.