The importance of the fine and normal filters, which are the theme of this work, appears naturally in the study of large cardinals. It is well known that the filter generated by the closed unbounded sets on Pκ(λ), where κ is a regular cardinal, is the minimal normal filter that contains the cones on each of these spaces. Henle and Zwicker  introduce the space Qκ(λ) of partitions of λ with size less than κ and described several notions of normality such that the existence of fine and “normal” ultrafilters on this space has strong implications, such as κ is λ supercompact. We propose in the present paper to study the existence of fine and “normal” filters on Qκ(λ). We will work with the different notions of normality introduced in  and also a different new definition for normality on Qκ(λ), characterizing, in case it exists, the minimal “normal” filter containing the cones on Qκ(λ).