This paper provides a comprehensive treatment of proper complex random variables and processes, highlighting their significance in statistical communication theory. Proper complex random variables and processes are characterized by vanishing pseudo-covariance, which ensures that their probability density function and differential entropy are fully specified by their mean and conventional covariance matrix. The paper demonstrates that the complex multivariate Gaussian density assumes its natural form only for proper random variables. It also shows that the differential entropy of a complex random vector with a fixed correlation matrix is maximized if and only if the vector is proper, Gaussian, and zero-mean. The concept of circular stationarity is introduced, and a discrete Fourier transform correspondence is derived for proper complex random processes, relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. An application of this correspondence simplifies the derivation of the capacity of a discrete-time Gaussian channel with memory, extending the results to channels with proper complex additive white Gaussian noise and a complex unit-sample response.This paper provides a comprehensive treatment of proper complex random variables and processes, highlighting their significance in statistical communication theory. Proper complex random variables and processes are characterized by vanishing pseudo-covariance, which ensures that their probability density function and differential entropy are fully specified by their mean and conventional covariance matrix. The paper demonstrates that the complex multivariate Gaussian density assumes its natural form only for proper random variables. It also shows that the differential entropy of a complex random vector with a fixed correlation matrix is maximized if and only if the vector is proper, Gaussian, and zero-mean. The concept of circular stationarity is introduced, and a discrete Fourier transform correspondence is derived for proper complex random processes, relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. An application of this correspondence simplifies the derivation of the capacity of a discrete-time Gaussian channel with memory, extending the results to channels with proper complex additive white Gaussian noise and a complex unit-sample response.