  Starting from a complete graph on $n$ vertices, repeatedly delete the edges
of a uniformly chosen triangle. This stochastic process terminates once it
arrives at a triangle-free graph, and the fundamental question is to estimate
the final number of edges (equivalently, the time it takes the process to
finish, or how many edge-disjoint triangles are packed via the random greedy
algorithm). Bollob\'as and Erd\H{o}s (1990) conjectured that the expected final
number of edges has order $n^{3/2}$, motivated by the study of the Ramsey
number $R(3,t)$. An upper bound of $o(n^2)$ was shown by Spencer (1995) and
independently by R\"odl and Thoma (1996). Several bounds were given for
variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald
(1999)), while the best known upper bound for the original question of
Bollob\'as and Erd\H{o}s was $n^{7/4+o(1)}$ due to Grable (1997). No nontrivial
lower bound was available.
  Here we prove that with high probability the final number of edges in random
triangle removal is equal to $n^{3/2+o(1)}$, thus confirming the 3/2 exponent
conjectured by Bollob\'as and Erd\H{o}s and matching the predictions of Spencer
et al. For the upper bound, for any fixed $\epsilon>0$ we construct a family of
$\exp(O(1/\epsilon))$ graphs by gluing $O(1/\epsilon)$ triangles sequentially
in a prescribed manner, and dynamically track all homomorphisms from them,
rooted at any two vertices, up to the point where $n^{3/2+\epsilon}$ edges
remain. A system of martingales establishes concentration for these random
variables around their analogous means in a random graph with corresponding
edge density, and a key role is played by the self-correcting nature of the
process. The lower bound builds on the estimates at that very point to show
that the process will typically terminate with at least $n^{3/2-o(1)}$ edges
left.
