Let $X$ be a completely regular space. We denote by $C_{b}(X)$ the Banach 
space of all real-valued bounded continuous functions on $X$ endowed with the 
supremun-norm.
In this paper we prove some characterisations of weakly compact operators 
from $C_{b}(X)$ into a Banach space $E$ which are continuous with respect to
$\beta_{t},\beta_{\sigma},\beta_{\tau},\beta_{s}$ and $\beta_{g}$, strict
topologies.
We also prove that $(C_{b}(X),\beta_{i});i=t,\tau,p,s,g$ has the 
Dunford-Pettis property.