A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if
for all non-increasing sequences of positive real numbers $\{a_n\}$ with
$\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set
\[\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}}
B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty
\text{many}n\geq1\}\] has full Lebesgue measure. (To put it informally, BC
sequences are sequences for which a natural converse to the Borel-Cantelli
Theorem holds).

This paper proves shrinking target results for IETs. Let {a_1\geq a_2
\geq...} be a sequence of positive real numbers with divergent sum. Then for
almost every IET T, the limsup of B(T^ix,a_i) has full Lebesgue measure (where
B(z, e) is the open ball around z of radius e). Related results are established
including the analogous result for geodesic flows on a translation surface.