Let ϕ be a real-valued valuation on the family of compact convex subsets of Rn and let K be a convex body in Rn. We introduce the ϕ-covariogram gK,ϕ of K as the function associating to each x∈Rn the value ϕ(K∩(K+x)). If ϕ is the volume, then gK,ϕ is the covariogram, extensively studied in various sources. When ϕ is a quermassintegral (e.g., surface area or mean width) gK,ϕ has been introduced by Nagel [26]. We study various properties of ϕ-covariograms, mostly in the case n=2 and under the assumption that ϕ is translation invariant, monotone, and even. We also consider the generalization of Matheron's covariogram problem to the case of ϕ-covariograms, that is, the problem of determining an unknown convex body K, up to translations and point reflections, by the knowledge of gK,ϕ. A positive solution to this problem is provided under different assumptions, including the case where K is a polygon and ϕ is either strictly monotone or ϕ is the width in a given direction. We prove that there are examples in every dimension n≥3 where K is determined by its covariogram but it is not determined by its width-covariogram. We also present some consequence of this study in stochastic geometry.