The properties (and particularly the superfluidity) of liquid helium near the $\lambda$ point have long been and still are objects of numerous investigations. Nonetheless, much of the problem remains unsolved from both the theoretical and the experimental points of view. The main reason is the small correlation length in helium II even when the distance from the $\lambda$ point is as small as hundredths of a degree, so that to reveal a number of specific effects one must work quite close to the $\lambda$ point. This article treats in detail the phenomenological theory of superfluidity of helium near the $\lambda$ point, a theory whose development dates back to 1958, and which is based on the use of the complex order parameter $\Psi=\eta e^\varphi$ (the density of the superfluid part of helium is here $\rho_s=m\eta^2$, and the velocity of this helium component is $v=(\hbar/m)\nabla\varphi$ where $m$ is the mass of the helium atom). In addition to formulating the general equations of the theory and discussing the regions where they are valid, we consider a number of concrete effects. The predictions of the theory are quite rich in content and can be verified in experiment. The main purpose of the article is just to contribute to such a verification.