The areas of Ramsey theory and random graphs have been closely linked ever since Erdos's famous proof in 1947 that the ``diagonal'' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the ``off-diagonal'' Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_n,triangle $. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that $R(3,k) = Theta big ( k^2 / log k big )$. In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.