Consider a real-valued branching random walk in the boundary case and denote by $\M_n$ its minimum at generation $n$. As $n \to \infty$, $\M_n- {3 \over 2} \log n$ is tight (see [1, 8, 2]). We establish here a law of iterated logarithm for the upper limits of $\M_n- {3\over 2} \log n$ and study the moderate deviation problem which is closely related to the small deviations of Mandelbrot's cascades.