  In this paper first we survey the Tur\'an type inequalities and related
problems for the Bessel functions of the first kind. Then we extend the known
higher order Tur\'an type inequalities for Bessel functions of the first kind
to real parameters and we deduce new closed integral representation formulae
for the second kind Neumann type series of Bessel functions of the first kind
occurring in the study of Tur\'an determinants of Bessel functions of the first
kind. At the end of the paper we prove a Tur\'an type inequality for the Bessel
functions of the second kind.
