The 3D Tune-In Toolkit (3DTI Toolkit) is an open-source standard C++ library which includes a binaural spatialiser. This paper presents the technical details of this renderer, outlining its architecture and describing the processes implemented in each of its components. In order to put this description into context, the basic concepts behind binaural spatialisation are reviewed through a chronology of research milestones in the field in the last 40 years. The 3DTI Toolkit renders the anechoic signal path by convolving sound sources with Head Related Impulse Responses (HRIRs), obtained by interpolating those extracted from a set that can be loaded from any file in a standard audio format. Interaural time differences are managed separately, in order to be able to customise the rendering according the head size of the listener, and to reduce comb-filtering when interpolating between different HRIRs. In addition, geometrical and frequency-dependent corrections for simulating near-field sources are included. Reverberation is computed separately using a virtual loudspeakers Ambisonic approach and convolution with Binaural Room Impulse Responses (BRIRs). In all these processes, special care has been put in avoiding audible artefacts produced by changes in gains and audio filters due to the movements of sources and of the listener. The 3DTI Toolkit performance, as well as some other relevant metrics such as non-linear distortion, are assessed and presented, followed by a comparison between the features offered by the 3DTI Toolkit and those found in other currently available open- and closed-source binaural renderers.