This paper describes the R package mvLSW. The package contains a suite of tools for the analysis of multivariate locally stationary wavelet (LSW) time series. Key elements include: (i) the synthesis of multivariate LSW time series for a given multivariate evolutionary wavelet spectrum (EWS); (ii) estimation of the time-dependent multivariate EWS for a given time series; (iii) estimation of the time-dependent coherence and partial coherence between time series channels; and, (iv) estimation of confidence intervals for the multivariate EWS estimation. A demonstration of the package is presented via both a simulated example and a case study using the EuStockMarkets data from the R data repository.