We address the problem of controlling the curvature of a Bézier curve interpolating a given set of data. More precisely, given two points $M$ and $N$, two directions $\vec{u}$ and $\vec{v}$, and a constant $k$, we would like to find two quadratic Bézier curves $\Gamma_1$ and $\Gamma_2$ joined with continuity $G^1$, interpolating the two points $M$ and $N$, such that the tangent vectors at $M$ and $N$ have directions $\vec{u}$ and $\vec{v}$ respectively, the curvature is everywhere upper bounded by $k$, and some evaluating function, the length of the resulting curve for example, is minimized. In order to solve this problem we need first to determine the maximum curvature of quadratic Bézier curves. This problem was solved by Sapidis and Frey in 1992. Here we present a simpler formula that has an elegant geometric interpretation in terms of distances and areas determined by the control points. We then use this formula to solve several problems. In particular, we solve the variant of the curvature control problem in which $\Gamma_1$ and $\Gamma_2$ are joined with continuity $C^1$, where the length $\alpha$ between the first two control points of $\Gamma_1$ is equal to the length between the last two control points of $\Gamma_2$, and where $\alpha$ is the evaluating function to be minimized. We also study the variant where we require a continuity $G^2$, instead of $C^1$, at the junction point. Finally, given two endpoints of a quadratic Bézier curve $\Gamma$, we characterize the locus of control points such that the maximum curvature of $\Gamma$ is prescribed.