We study the boundedness of Toeplitz operators $T_a$ with locally
integrable symbols on Bergman spaces $A^p(\mathbb{D})$,
$1 < p < \infty$.
Our main result gives a sufficient condition for the boundedness
of $T_a$ in terms of some ``averages'' (related to hyperbolic
rectangles) of its symbol. If the averages satisfy an ${o}$-type
condition on the boundary of $\mathbb{D}$, we show that the
corresponding Toeplitz operator is compact on $A^p$. Both
conditions coincide with the known necessary conditions in the
case of nonnegative symbols and $p=2$. We also show that Toeplitz
operators with symbols of vanishing mean oscillation are Fredholm
on $A^p$ provided that the averages are bounded away from zero,
and derive an index formula for these operators.