In this thesis we study the boundedness of a generalization of the Hardy-Littlewood maximal operator, involving rearrangement invariant Banach function space and indices of the spaces. We first consider a classical proof of boundedness of the Hardy-Littlewood maximal operator on rearrangement invariant Banach function spaces. After establishing necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, we consider a generalization of the Hardy-Littlewood maximal operator introduced by C. Pérez. We investigate and slightly improve the known sufficient conditions under which this more general maximal operator is bounded on a rearrangement invariant Banach function space. After which we search and find necessary conditions for boundedness in a general setting. In the final section we study Boyd indices and fundamental indices, especially how they are related to boundedness of the more general maximal operator. We also introduce weak fundamental indices and investigate some of their properties and uses. Finally we show how under certain assumptions we can state equivalent necessary and sufficient conditions for boundedness on Lorentz spaces and Orlicz spaces.