We present a systematic study of exclusive charmless nonleptonic two-body $B$ decays including axial-vector mesons in the final state. We calculate branching ratios of $B\ensuremath{\rightarrow}PA$, $VA$, and $AA$ decays, where $A$, $V$, and $P$ denote an axial vector, a vector, and a pseudoscalar meson, respectively. We assume a naive factorization hypothesis and use the improved version of the nonrelativistic Isgur-Scora-Grinstein-Wise quark model for form factors in $B\ensuremath{\rightarrow}A$ transitions. We include contributions that arise from the effective $\ensuremath{\Delta}B=1$ weak Hamiltonian ${H}_{\mathrm{eff}}$. The respective factorized amplitudes of these decays are explicitly shown and their penguin contributions are classified. We find that decays ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{a}_{1}^{0}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${\overline{B}}^{0}\ensuremath{\rightarrow}{a}_{1}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}}$, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{a}_{1}^{\ensuremath{-}}{\overline{K}}^{0}$, ${\overline{B}}^{0}\ensuremath{\rightarrow}{a}_{1}^{+}{K}^{\ensuremath{-}}$, ${\overline{B}}^{0}\ensuremath{\rightarrow}{f}_{1}{\overline{K}}^{0}$, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{f}_{1}{K}^{\ensuremath{-}}$, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{K}_{1}^{\ensuremath{-}}(1400){\ensuremath{\eta}}^{(\ensuremath{'})}$, ${B}^{\ensuremath{-}}\ensuremath{\rightarrow}{b}_{1}^{\ensuremath{-}}{\overline{K}}^{0}$, and ${\overline{B}}^{0}\ensuremath{\rightarrow}{b}_{1}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}({K}^{\ensuremath{-}})$ have branching ratios of the order of ${10}^{\ensuremath{-}5}$. We also study the dependence of branching ratios for $B\ensuremath{\rightarrow}{K}_{1}P(V,A)$ decays [${K}_{1}={K}_{1}(1270)$, ${K}_{1}(1400)$] with respect to the mixing angle between ${K}_{1A}$ and ${K}_{1B}$.