In this paper, the authors defined a multifunction $F\colon X\to Y$ to be upper (resp. lower) weakly $\alpha$-continuous if for each $x\in X$ and each open set $V$ of $Y$ such that $F(x)\subset V$ (resp. $F(x)\cap V\neq\emptyset$), there exists an $\alpha$-open set $U$ of $X$ containing $x$ such that $U\subset F^+(\mathrm{Cl}(V))$ (resp. $U\subset F^-(\mathrm{Cl}(V))$). They give some characterizations and several properties concerning upper (lower) weakly $\alpha$-continuous multifunctions.