NumSBT: A subroutine for calculating spherical Bessel transforms numerically. A previous subroutine, LSFBTR, for computing numerical spherical Bessel (Hankel) transforms is updated with several improvements and modifications. The procedure is applicable if the input radial function and the output transform are defined on logarithmic meshes and if the input function satisfies reasonable smoothness conditions. Important aspects of the procedure are that it is simply implemented with two successive applications of the fast Fourier transform, and it yields accurate results at very large values of the transform variable. Applications to the evaluation of overlap integrals and the Coulomb potential of multipolar charge distributions are described.