  In this paper we begin the first systematic study of distributions of simple
marked mesh patterns. Mesh patterns were introduced recently by Br\"and\'en and
Claesson in connection with permutation statistics. We provide explicit
generating functions in several general cases, and develop recursions to
compute the numbers in question in some other cases. Certain $q$-analogues are
discussed. Moreover, we consider two modifications of the notion of a marked
mesh pattern and provide enumerative results for them.
