The Minkowski problem forms the core of various areas in fully nonlinear partial differential equations on the sphere and convex geometry, and is intimately connected to the Brunn-Minkowski inequality about convex bodies. Recently Lutwak's $L_p$-Minkowski problem and its variants have received significant attention where the $p=1$ case corresponds to the classical Minkowski problem, and the case $p=0$ is the so-called Logarithmic Minkowski problem going back to Firey. The talk surveys developments about the $L_p$-Minkowski problem and the central Logarithmic Minkowski Conjecture concerning the uniqueness of the even solution in the $p=0$ case, and the relation of this conjecture to some strengthening of the Brunn-Minkowski inequality for origin symmetric convex bodies, to some inequalities for the Gaussian density, and to some spectral gap estimates for certain self-adjoint operators.