Let $G$ and $H$ be Archimedean Riesz spaces. We study the properties of band operators and inverse band operators from $G$ to $H$ and investigate their relations to some well-known classes of operators. Then, we show that under some assumptions on the Riesz spaces $G$ or $H$, if $S$ is a bijective band operator from $G$ into $H$ then $S^{-1}:H\rightarrow G$ is a band operator. Additionally, we give some conditions under which a band operator becomes order bounded.