In this chapter we introduce some of the key methodological concepts that will be used extensively throughout the book. These methods include dimension reduction using principal components analysis (PCA) and singular value decomposition (SVD), Gaussian Processes, semiparametric smoothing using regression and penalized splines, the mixed effects representation of penalized splines models, correlation and multiplicity adjusted inference (confidence intervals and p-values), and covariance smoothing for dense and sparse functional data. These concepts will be used by themselves or together to address many methodological and computational challenges of modeling functional data analysis.