||There are two standard approaches to the construction of t-designs. The first one is based on permutation group actions on certain base blocks. The second one is based on coding theory. The objective of this paper is to give a spectral characterisation of all t-designs by introducing a characteristic Boolean function of a t-design. The spectra of the characteristic functions of (n 2)/2(n, n/2, 1) Steiner systems are determined and properties of such designs are proved. Delsarte's characterisations of orthogonal arrays and t-designs, which are two special cases of Delsarte's characterisation of T-designs in association schemes, are slightly extended into two spectral characterisations. Another characterisation of t-designs by Delsarte and Seidel is also extended into a spectral one. These spectral characterisations are then compared with the new spectral characterisation of this paper.