In this paper, we introduce plane permutations, i.e. pairs $mathfrak{p}=(s,pi)$ where $s$ is an $n$-cycle and $pi$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $pi$) and the diagonal ($D_{mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plane permutations obtained by permuting their diagonal-blocks. We establish basic properties of plane permutations and study transpositions and exceedances and derive various enumerative results. In particular, we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing Chapuys recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of $n$-cycles $omega$, for which the product $omega(1~2~cdots ~n)$ has exactly $k$ cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.
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