  Geometric properties of the Jackson and Hahn-Exton $q$-Bessel functions are
studied. For each of them, three different normalizations are applied in such a
way that the resulting functions are analytic in the unit disk of the complex
plane. For each of the six functions we determine the radii of starlikeness and
convexity precisely by using their Hadamard factorization. These are
$q$-generalizations of some known results for Bessel functions of the first
kind. The characterization of entire functions from the Laguerre-P\'olya class
via hyperbolic polynomials play an important role in this paper. Moreover, the
interlacing property of the zeros of Jackson and Hahn-Exton $q$-Bessel
functions and their derivatives is also useful in the proof of the main
results. We also deduce a sufficient and necessary condition for the
close-to-convexity of a normalized Jackson $q$-Bessel function and its
derivatives. Some open problems are proposed at the end of the paper.
