The notion of mesh patterns in permutations was introduced recently by Petter Branden and Anders Claesson to provide explicit expansions for certain permutation statistics as possibly infinite linear combinations of (classical) permutation patterns. In my talk, I will discuss eight mesh patterns of small lengths. In particular, I will link avoidance of one of the patterns to the harmonic numbers, while for three other patterns I will show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, I will show that two specific mesh patterns are Wilf equivalent meaning that for any length, the number of permutations avoiding one of the patterns equals that avoiding the other one. As a byproduct of these studies, one defines a new set of sequences counted by the Catalan numbers and provides a relation on the Catalan triangle that seems to be new. This is joint work with Jeffrey Liese.