Score-based diffusion models have recently emerged as state-of-the-art generativemodels for a variety of data modalities. Nonetheless, it remains unclear how toadapt these models to generate long multivariate time series. Viewing a timeseries as the discretisation of an underlying continuous process, we introduceSigDiffusion, a novel diffusion model operating on log-signature embeddingsof the data. The forward and backward processes gradually perturb and denoiselog-signatures while preserving their algebraic structure. To recover a signal fromits log-signature, we provide new closed-form inversion formulae expressing thecoefficients obtained by expanding the signal in a given basis (e.g. Fourier ororthogonal polynomials) as explicit polynomial functions of the log-signature.Finally, we show that combining SigDiffusions with these inversion formulaeresults in high-quality long time series generation, competitive with the currentstate-of-the-art on various datasets of synthetic and real-world examples.