  We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple
$C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups
${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$.
K-theoretical obstruction enables us to compute the fundamental group easily.
Our study is essentially based on the computation of Picard groups by Kodaka.
