The covariogram of a convex body $K$ provides the volumes of the intersections of $K$ with all its possible translates. Matheron conjectured in 1986 that this information determines $K$ among all convex bodies, up to certain known ambiguities. It is proved that this is the case if $K\subset{\mathbb R}^2$ is not ${\rm C}^1$, or it is not strictly convex, or its boundary contains two arbitrarily small ${\rm C}^2$ open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in ${\mathbb R}^n$ for any $n\geq 4$.