The authors address the problem of likelihood based inference for correlated diffusions. Such a task presents two issues; the positive definite constraints of the diffusion matrix and the likelihood intractability. The first issue is handled by using the Cholesky factorisation on the diffusion matrix. To deal with the likelihood unavailability, a generalisation of the data augmentation framework of Roberts and Stramer (2001 Biometrika 88(3), 603-621) to d-dimensional correlated diffusions, including multivariate stochastic volatility models, is given. The methodology is illustrated through simulated and real datasets.