We establish an Azuma inequality for martingales in the noncommutative setting and apply it to deduce a noncommutative Heoffding inequality. We prove that if $\mathfrak{M}$ is a von Neumann algebra with unit element $1$ equipped with a faithful normal finite trace $\tau$ such that $\tau(1)=1$, $x=(x_j)_{0\leq j\leq n}$ is a self-adjoint martingale with respect to a filtration $(\mathfrak{M}_n, \mathcal{E}_n)_{n\geq 0}$ of von Neumann subalgebras of $\mathfrak{M}$ and $dx_j=x_j-x_{j-1}$ is its associated martingale differences satisfying $\mathcal{E}_{j-1}((dx_j)^2)\leq\sigma_j^2$ and $dx_j\leq a_j+M$ for some positive constants $a_j, \sigma_j$ and $M$ and all $1\leq j\leq n$, then \begin{eqnarray*} {\rm Prob}\left(\left|\sum_{j=1}^ndx_j\right|\geq \lambda\right)\leq 2\exp\left\{\frac{-\lambda^2}{2\left(\sum_{j=1}^n(\sigma_j^2+a_j^2\right)+M\lambda/3)}\right\}. \end{eqnarray*} for all $\lambda > 0$. Further, we obtain several consequences.

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