Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This thesis synthesizes descriptions of kernel methods and operator theory into a single consistent account of KAF, one that illuminates the property that, under measure-preserving and ergodic dynamics, KAF consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical system, conditioned on the observed data at forecast initialization. More precisely, KAF yields optimal predictions, in the sense of minimal root mean square error with respect to the invariant measure, in the asymptotic limit of large data. The presented framework facilitates the analysis of generalization error, the quantification of uncertainty, and the calculation of conditional variance and conditional probability. Applications of KAF to synthetic dynamical systems, namely periodic flow on the unit circle and the chaotic Lorenz-63 system, are presented for illustrative purposes, and the viability of KAF in real-world operational settings is explored through applications to tropical atmospheric dynamics. Applied to satellite-obtained global brightness temperature data, KAF produces forecasts of the Madden Julian oscillation (MJO) and boreal summer intraseasonal oscillation (BSISO) with pattern correlations above 0.6 for lead times of up to 50 days when 23 years of training data are used, and up to 37 days for the MJO when 9 years of data are used. Applied to satellite and ground station observed precipitation data, KAF produces forecasts of the Indian Summer Monsoon Rainfall (ISMR) that remain skillful for a lead time of up to 10 days (2 pentads) when 20 years of training data are used.