Summary: Let \(\mathcal{F}\) be a family of convex sets in \(\mathbb{R}^d,\) which are colored with \(d + 1\) colors. We say that \(\mathcal{F}\) satisfies the Colorful Helly Property if every rainbow selection of \(d + 1\) sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family \(\mathcal{F}\) there is a color class \(\mathcal{F}_i \subset \mathcal{F},\) for \(1 \le i \le d + 1,\) whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension \(d \ge 2\) there exist numbers \(f(d)\) and \(g(d)\) with the following property: either one can find an additional color class whose sets can be pierced by \(f(d)\) points, or all the sets in \(\mathcal{F}\) can be crossed by \(g(d)\) lines.