Let K be a convex body in R^n with its centroid at the origin, let u be a unit vector, and let u^+ be the n-dimensional half-space which contains u and whose boundary is orthogonal to u. Grunbaum’s inequality bounds the volume of K \cap u^+ from below by 1/e times the volume of K. I will discuss several recent extensions of this inequality, including our joint work with Ning Zhang where we establish a "Grunbaum’s inequality for projections" and a "Grunbaum’s inequality for sections".