Truncated Poisson operator is an example of one-dimensional convolution integral equation on an interval with a very simple kernel which evades applicability of most of known constructive technics. After discussing general spectral structure and basic properties of eigenfunctions, I present newly developed methods for their asymptotic approximations in regimes of the small and large interval (or, alternatively, the parameter of the Poisson kernel). I point out connections with prolate spheroidal wave functions and spectrum of the simplest hypersingular integral operator, and then explain non-trivial asymptotic reducibility of the problem to an equation of Wiener-Hopf type amenable to explicit solution. Asymptotic analysis leads to nearly trigonometric structure of eigenfunctions which is confirmed by numerical simulations. Finally, I briefly mention some applications and variety of contexts where equations with Poisson kernel arise under names of Love, Gaudin and Lieb-Liniger.