We introduce the notion of tracial topologicalrank for ${\rm C}^*$-algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of ${\rm C}^*$-algebrasof the form $PM_n(C(X))P$, where $X$ is a compact metric space with ${\rm dim\,} X\le k$, and$P$ is a projection in $M_n(C(X))$,have tracial topological rank no more than $k$.Non-nuclear ${\rm C}^*$-algebras can have small tracial topological rank. It is shown that if $A$ is a simple unital${\rm C}^*$-algebra with tracial topological rank $k$ ($<\infty$), then\begin{enumerate} \item[(i)] $A$ is quasidiagonal,\item[(ii)] $A$ has stable rank $1$,\item[(iii)] $A$ has weakly unperforated $K_0(A)$,\item[(iv)] $A$ has the following Fundamental Comparability of Blackadar: if $p,q\in A$ are two projections with$\tau(p)<\tau(q)$ for all tracial states $\tau$on $A$, then $p\preceq q$. 2000 Mathematics Subject Classification:46L05, 46L35.
