This paper introduces the concept of proper complex random variables and processes, which are characterized by a vanishing pseudo-covariance. Properness is preserved under affine transformations and is essential for the complex-multivariate Gaussian density to take a natural form. The maximum-entropy theorem is generalized to the complex case, showing that the differential entropy of a complex random vector with a fixed correlation matrix is maximized when the vector is zero-mean Gaussian and proper. Circular stationarity is introduced, and a discrete Fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. The capacity of a discrete-time channel with complex inputs, proper complex additive white Gaussian noise, and a finite complex unit-sample response is determined, simplifying earlier derivations for real channels. The results are applied to show that the capacity of a complex discrete-time Gaussian channel with memory is achieved when the input is proper, Gaussian, and c.w.s.s. with zero mean. The paper also demonstrates that the real NCGC capacity is half that of the complex NCGC, with the real input sequence being Gaussian and c.w.s.s. with zero mean.This paper introduces the concept of proper complex random variables and processes, which are characterized by a vanishing pseudo-covariance. Properness is preserved under affine transformations and is essential for the complex-multivariate Gaussian density to take a natural form. The maximum-entropy theorem is generalized to the complex case, showing that the differential entropy of a complex random vector with a fixed correlation matrix is maximized when the vector is zero-mean Gaussian and proper. Circular stationarity is introduced, and a discrete Fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain. The capacity of a discrete-time channel with complex inputs, proper complex additive white Gaussian noise, and a finite complex unit-sample response is determined, simplifying earlier derivations for real channels. The results are applied to show that the capacity of a complex discrete-time Gaussian channel with memory is achieved when the input is proper, Gaussian, and c.w.s.s. with zero mean. The paper also demonstrates that the real NCGC capacity is half that of the complex NCGC, with the real input sequence being Gaussian and c.w.s.s. with zero mean.