We propose the use of higher order tensors, and their decompositions, for efficient analysis of combustion direct numerical simulation (DNS) data. Turbulent combustion DNS data, being inherently multiscale and multivariate, pose many challenges and higher order tensors are a natural abstraction to organise, probe and analyse them. The chapter gives a high-level overview of prominent tensor decomposition methods, their interpretation, algorithmic challenges and desirable properties. Two examples of DNS analysis employing tensor decompositions are then presented. The first analysis, based on truncated higher order singular value decomposition (truncated HOSVD), also known as Tucker decomposition, allows significant, albeit lossy, compression of DNS data, which may be inevitable in the exascale computing era. The factors aiding, and impeding, compression and the implications in terms of element-wise error distributions are presented using three candidate DNS data sets. The second analysis is centred on higher order joint moment tensors, which are richly informative for multivariate non-Gaussian variables. An anomaly detection algorithm based on the decomposition of the fourth moment tensor is presented, and its ability in detecting localised auto-ignition kernels in a homogeneous charge compression ignition (HCCI) data set is examined.