Summary: In this paper, we introduce an \(m\)-fold illumination number \(I^m(K)\) of a convex body \(K\) in Euclidean space \(\mathbb{E}^d\), which is the smallest number of directions required to \(m\)-fold illuminate \(K\), i.e., each point on the boundary of \(K\) is illuminated by at least \(m\) directions. We get a lower bound of \(I^m(K)\) for any \(d\)-dimensional convex body \(K\), and get an upper bound of \(I^m(K)\) for any \(d\)-dimensional convex body \(K\) with smooth boundary. We also prove that \(I^m(K)=2m+1\), for a 2-dimensional smooth convex body \(K\). Furthermore, we obtain some results related to the \(m\)-fold illumination numbers of convex polygons and cap bodies of a \(d\)-dimensional unit ball \(\mathbb{B}^d\) in small dimensions. In particular, we show that \(I^m (P) = \lceil mn / \lfloor\frac{n-1}{2}\rfloor\rceil\), for a regular convex \(n\)-sided polygon \(P\).