Abstract: In this first lecture, we will make the assumption that the full curvature tensor is bounded, say $|R|\leq 1$. The key distinction is between the not too noncollapsed case, in which a sufficiently small ball can be shown to look like a ball in $R^n$, and the sufficiently collapsed case, in which it turns out that $B_{\epsilon(n)}(m)$ looks like an $\epsilon(n)$-tubular neighborhood of the zero section of a vector bundle over some (tiny) nilmanifold. If $B_1(m)$ is sufficiently collapsed for all $m\in M^n$, a coherent collection of nilmanifolds arises, encoding the essential features of the collapsed geometry.