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Description: A wide variety of methods are available to estimate the between-study variance under the univariate random-effects model for meta-analysis. Some, but not all, of these estimators have been extended so that they can be used in the multivariate setting. We begin by extending the univariate generalised method of moments, which immediately provides a wider class of multivariate methods than was previously available. However, our main proposal is to use this new type of estimator to derive multivariate multi-step estimators of the between-study covariance matrix. We then use the connection between the univariate multi-step and Paule-Mandel estimators to motivate taking the limit, where the number of steps tends towards infinity. We illustrate our methodology using two contrasting examples and investigate its properties in a simulation study. We conclude that the proposed methodology is a fully viable alternative to existing estimation methods. Advantages of the new multivariate multi-step estimator include its semi-parametric nature and that it is computationally feasible in high dimensions. Our proposed estimation methods are also applicable for multivariate random-effects meta-regression, where study level covariates are included in the model.