The classical $ell^$ sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces $ell^_$ of analytic functions whose Taylor coefficients belong to $ell^p$. Relations between the Banach space $ell^p$ and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James...