  Permutation tableaux are combinatorial objects related with permutations and
various statistics on them. They appeared in connection with total positivity
in Grassmannians, and stationary probabilities in a PASEP model. In particular
they gave rise to an interesting q-analog of Eulerian numbers. The purpose of
this article is to study some combinatorial properties of type B permutation
tableaux, defined by Lam and Williams, and links with signed permutation
statistics. We show that many of the tools used for permutation tableaux
generalize in this case, including: the Matrix Ansatz (a method originally
related with the PASEP), bijections with labeled paths and links with continued
fractions, bijections with signed permutations. In particular we obtain a
q-analog of the type B Eulerian numbers, having a lot in common with the
previously known q-Eulerian numbers: for example they have a nice symmetry
property, they have the type B Narayana numbers as constant terms. The signed
permutation statistics arising here are of several kinds. Firstly, there are
several variants of descents and excedances, and more precisely of flag
descents and flag excedances. Other statistics are the crossings and
alignments, which generalize a previous definition on (unsigned) permutations.
There are also some pattern-like statistics arising from variants of the
bijection of Fran\ccon and Viennot.
