The Burgers equation in the critical case \(\alpha=1\), i.e., \[ \left. \begin{aligned} &\partial_t u + u\partial_x u + \Lambda u = 0,\\ &u(x,0)=u_0(x), \end{aligned}\right\} \quad x\in\mathbb{R}, \] is studied in the context of the Besov space \(\dot{B}^{1/p}_{p,1}(\mathbb{R})\), \(p\in [1,\infty)\). Here \(\Lambda\) is defined via \(\mathcal{F}(\Lambda^\alpha u)(\xi) = |\xi|^\alpha \mathcal{F}u(\xi)\), \(0\leq \alpha\leq 2\). This is closely related to recent similar results in the periodic case in the Hilbert space setting \(H^{1/2}(\mathbb{T}^1)\) in [A. Kiselev, F. Nazarov and R. Shterenberg, Dyn. Partial Differ. Equ. 5, No. 3, 211–240 (2008; Zbl 1186.35020)]. Moreover, the authors use methods and arguments developed there and in [H. Abidi and T. Hmidi, SIAM J. Math. Anal. 40, No. 1, 167–185 (2008; Zbl 1157.76054)]. The first main result proves uniqueness of the global solution \(u\) of the above PDE with regularity \[ u\in \mathcal{C}(\mathbb{R}^+; \dot{B}^{\frac1p}_{p,1})\cap L^1_{\mathrm{loc}}(\mathbb{R}^+; \dot{B}^{\frac1p+1}_{p,1}), \] assuming that the initial value satisfies \(u_0\in \dot{B}^{1/p}_{p,1}(\mathbb{R})\), \(p\in [1,\infty)\). The major part of the paper is devoted to the proof of the local well-posedness which itself requires an optimal a priori estimate for the transport-diffusion equation in \(\mathbb{R}^N\), \[ \begin{cases} \partial_t u + v \cdot \nabla u + \nu \Lambda^\alpha u = f,\\ u(x,0)=u_0(x), \end{cases} \] where \(v\) is a given vector field (not necessarily divergence free), \(f\) a given external force, \(\nu\geq 0\), and \(0\leq\alpha\leq 2\). The precise formulation is contained in Theorem 1.2 and was known in case of \(\alpha=2\) before. The authors extend the result in [R. Danchin, J. Hyperbolic Differ. Equ. 4, No. 1, 1-17 (2007; Zbl 1117.35012)] to the general situation \(\alpha\in [0,2]\).