Many sources of fluctuation contribute to the fMRI signal, and this makes identifying the effects that are truly related to the underlying neuronal activity difficult. Independent component analysis (ICA) - one of the most widely used techniques for the exploratory analysis of fMRI data - has shown to be a powerful technique in identifying various sources of neuronally-related and artefactual fluctuation in fMRI data (both with the application of external stimuli and with the subject "at rest"). ICA decomposes fMRI data into patterns of activity (a set of spatial maps and their corresponding time series) that are statistically independent and add linearly to explain voxel-wise time series. Given the set of ICA components, if the components representing "signal" (brain activity) can be distinguished form the "noise" components (effects of motion, non-neuronal physiology, scanner artefacts and other nuisance sources), the latter can then be removed from the data, providing an effective cleanup of structured noise. Manual classification of components is labour intensive and requires expertise; hence, a fully automatic noise detection algorithm that can reliably detect various types of noise sources (in both task and resting fMRI) is desirable. In this paper, we introduce FIX ("FMRIB's ICA-based X-noiseifier"), which provides an automatic solution for denoising fMRI data via accurate classification of ICA components. For each ICA component FIX generates a large number of distinct spatial and temporal features, each describing a different aspect of the data (e.g., what proportion of temporal fluctuations are at high frequencies). The set of features is then fed into a multi-level classifier (built around several different classifiers). Once trained through the hand-classification of a sufficient number of training datasets, the classifier can then automatically classify new datasets. The noise components can then be subtracted from (or regressed out of) the original data, to provide automated cleanup. On conventional resting-state fMRI (rfMRI) single-run datasets, FIX achieved about 95% overall accuracy. On high-quality rfMRI data from the Human Connectome Project, FIX achieves over 99% classification accuracy, and as a result is being used in the default rfMRI processing pipeline for generating HCP connectomes. FIX is publicly available as a plugin for FSL.