This thesis is concerned with partial orders on set and integer partitions and relatedrnstructures. The study of set and integer partitions dates back to Euler and Sylvesterrn[28]. Over the course of time it has become apparent that the combinatorics of partitionsrnencodes important mathematical structures in a variety of felds. Results onrnpartially ordered sets of partitions involving Mfobius numbers, homology and homotopyrnof order complexes emerged in the works of Rota and Stanley [56]. In this work,rnwe use pointed partition structures where a part of an integer and set partition isrnmarked. These have been introduced in the works of Ehrenborg and Readdy [60],rnEhrenborg and Jung [63] and Ziegler [29]. We exhibit new enumerative and geometricrnproperties of the partial orders of pointed partitions studied by Ehrenborg andrnReaddy [60]. In particular, we compute the Mfobius numbers and homotopy types ofrnlower intervals.rnIn addition, we investigate their ordered counterparts and provide analogous resultsrnin this setting. Then we use the pointed set partition lattice to introduce exponentialrnpointed structures which are the pointed analog of exponential structures introducedrnby Stanley [55]. We show that this concept encompasses many examples introducedrnbefore. In particular, we introduce pointed decompositions of lattices and study theirrnenumerative and geometric structure. We also show that exponential pointed structuresrnsatisfy pointed analogs of Stanley's compositional and exponential formulas