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Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be thecoefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using theCaldero-Chapoton map, we introduce and investigate a family of genericvariables in $\Z[\textbf u^{\pm 1}]$ containing the cluster monomials of$\mathcal A(Q)$. The aim of these generic variables is to give an explicit newmethod for constructing $\Z$-bases in the cluster algebra $\mathcal A(Q)$.  If $Q$ is an affine quiver with minimal imaginary root $\delta$, weinvestigate differences between cluster characters associated to indecomposablerepresentations of dimension vector $\delta$. We define the notion of\emph{difference property} which gives an explicit description of thesedifferences. We prove in particular that this property holds for quivers ofaffine type $\tilde A$. When $Q$ satisfies the difference property, we provethat generic variables span the cluster algebra $\mathcal A(Q)$. If $\mathcalA(Q)$ satisfies some gradability condition, we prove that generic variables arelinearly independent over $\mathbb Z$ in $\mathcal A(Q)$. In particular, thisimplies that generic variables form a $\Z$-basis in a cluster algebraassociated to an affine quiver of type $\tilde A$.

Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras