  We show that every Lie algebroid $A$ over a manifold $P$ has a natural
representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top}
T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the
orientation bundle in topology, and sections of $Q_A$ may be viewed as
transverse measures to $A$. As a consequence, there is a well-defined class in
the first Lie algebroid cohomology $H^1(A)$ called the modular class of the Lie
algebroid $A$. This is the same as the one introduced earlier by Weinstein
using the Poisson structure on $A^*$. We show that there is a natural pairing
between the Lie algebroid cohomology spaces of $A$ with trivial coefficients
and with coefficients in $Q_A$. This generalizes the pairing used in the
Poincare duality of finite-dimensional Lie algebra cohomology. The case of
holomorphic Lie algebroids is also discussed, where the existence of the
modular class is connected with the Chern class of the line bundle $Q_A$.
