  A systematic study of avoidance of mesh patterns of length 2 was conducted by
Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this
paper, we give 27 distribution results for these patterns including 14
distributions for which avoidance was not known. Moreover, for the unsolved
cases, we prove an equidistribution result (out of 6 equidistribution results
we prove in total), and conjecture 6 more equidistributions. Finally, we find
seemingly unknown distribution of the well known permutation statistic ``strict
fixed point'', which plays a key role in many of our enumerative results.
  This paper is the first systematic study of distributions of mesh patterns.
Our techniques to obtain the results include, but are not limited to, obtaining
functional relations for generating functions, and finding recurrence relations
and bijections.
