We present a review of new results which suggest the existence of fully stable spinning solitons (self-supporting localised objects with an internal vorticity) in optical fibres with selffocusing Kerr (cubic) nonlinearity, and in bulk media featuring a combination of the cubic selfdefocusing and quadratic nonlinearities. Their distinctive difference from other optical solitons with an internal vorticity, which were recently studied in various optical media, theoretically and also experimentally, is that all the spinning solitons considered thus far have been found to be unstable against azimuthal perturbations. In the first part of the paper, we consider solitons in a nonlinear optical fibre in a region of parameters where the fibre carries exactly two distinct modes, viz., the fundamental one and the first-order helical mode. From the viewpoint of application to communication systems, this opens the way to doubling the number of channels carried by a fibre. Besides that, these solitons are objects of fundamental interest. To fully examine their stability, it is crucially important to consider collisions between them, and their collisions with fundamental solitons, in (ordinary or hollow) optical fibres. We introduce a system of coupled nonlinear Schr¨ odinger equations for the fundamental and helical modes with nonstandard values of the cross-phase-modulation coupling constants, and show, in analytical and numerical forms, results of collisions between solitons carried by the two modes. In the second part of the paper, we demonstrate that the interaction of the fundamental beam with its second harmonic in bulk media, in the presence of self-defocusing Kerr nonlinearity, gives rise to the first ever example of completely stable spatial ring-shaped solitons with intrinsic vorticity. The stability is demonstrated both by direct simulations and by analysis of linearized equations.