  We study the Carleson measures and the Toeplitz operators on the class of
so-called small weighted Bergman spaces, introduced recently by Seip. A
characterization of Carleson measures is obtained which extends Seip's results
from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use
this characterization to give necessary and sufficient conditions for the
boundedness and compactness of Toeplitz operators. Finally, we study the
Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.
