The main purpose of this work is to create a type of topological spaces namely "almost star α-Hurewicz spaces" and study its properties, and besides, the concepts of α-compact space, α-Hurewicz space, star α- Hurewicz space and strongly star α-Hurewicz space. Many properties of α-Hurewicz spaces and almost α-Hurewicz are investigated. This allows us to provide new examples of explicit descriptions of spaces as well as some types of α-covering for spaces such as α-compact and α-Lindelof space and using as a tool to prove important results in topological spaces. In addition, a certain connection of α-Hurewicz space with the Hurewicz space and almost α-Hurewicz space was considered. There is a relationship between the version of the strongly star α-Hurewicz property and star α-Hurewicz property with star α-compact property and almost star α-Hurewicz. Some of the examples that make a distinction between the properties are mentioned and reviewed, showing that the concepts are not equivalent. Results on the preservation of the properties of α-Hurewicz and almost-α-Hurewicz spaces are included such as behavior under subspaces, products, α-irresolute function, and mappings with other forms of topological spaces.