In this dissertation, we aim to answer three questions concerning group analysis of functional magnetic resonance imaging (fMRI) data. First, we propose a model-free cluster method that groups brain signals in to clusters, based on wavelet transformation and principal component analysis. From the clustered maps we identify activated regions related to the given tasks. We then use a resampling approach to compare clustered maps between practice groups and scan sessions. Second, we compare differences of groups of subjects in brain activation changes across two scan sessions. Using the property that brain signals in regions of interest (ROIs) may contain a similar pattern across subjects in a task-related experiment, we develop a semiparametric approach under shape invariance to quantify and test differences between sessions and groups. We conduct statistical inference on the scale parameter in the model to determine whether attenuation is present between two sessions for each group and whether a group difference exists between two sessions in multiple ROIs. Last, we take a functional data analysis approach to classify fMRI data. We propose a spatially weighted functional support vector machine (FSVM) that utilizes a parameter to estimate correlation between different brain regions. Using both numerical study and a real fMRI dataset, we show our proposed method achieves higher classification accuracy compared to a regular FSVM method.