We develop a general theory of continuous substitutions on compact Hausdorff alphabets. Focussing on implications of primitivity, we provide a self-contained introduction to the topological dynamics of their subshifts. We then reframe questions from ergodic theory in terms of spectral properties of the corresponding substitution operator. The standard Perron–Frobenius theory in finite dimensions is no longer applicable. To overcome this, we exploit the theory of positive operators on Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator and hence unique ergodicity of the associated subshift.