  We consider playing the game of Tic-Tac-Toe on block designs BIBD($v, k,
\lambda$) and transversal designs TD($k, n$). Players take turns choosing
points and the first player to complete a block wins the game. We show that
triple systems, BIBD($v, 3, \lambda$), are a first player win if and only if $v
\geq 5$. Further, we show that for $k = 2, 3$, TD($k, n$) is a first player win
if and only if $n \geq k$. We also consider a weak version of the game, called
Maker-Breaker, in which the second player wins if they can stop the first
player from winning. In this case, we adapt known bounds for when either the
first or second player can win on BIBD($v, k, 1$) and TD($k, n$), and show that
for Maker-Breaker, BIBD($v, 4, 1$) is a first player win if and only if $v \geq
16$. We show that TD($4, 4$) is a second player win, and so the second player
can force a draw in the regular game by playing the same strategy.
