The combination of functional magnetic resonance imaging (FMRI) and electroencephalography (EEG) has received much recent attention, since it potentially offers a new tool for neuroscientists that makes simultaneous use of the strengths of the two modalities. However, EEG data collected in such experiments suffer from two kinds of artifact. First, gradient artifacts are caused by the switching of magnetic gradients during FMRI. Second, ballistocardiographic (BCG) artifacts related to cardiac activities further contaminate the EEG data. Here we present new methods to remove both kinds of artifact. The methods are based primarily on the idea that temporal variations in the artifacts can be captured by performing temporal principal component analysis (PCA), which leads to the identification of a set of basis functions which describe the temporal variations in the artifacts. These basis functions are then fitted to, and subtracted from, EEG data to produce artifact-free results. In addition, we also describe a robust algorithm for the accurate detection of heart beat peaks from poor quality electrocardiographic (ECG) data that are collected for the purpose of BCG artifact removal. The methods are tested and are shown to give superior results to existing methods. The methods also demonstrate the feasibility of simultaneous EEG/FMRI experiments using the relatively low EEG sampling frequency of 2048 Hz.