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An edge-coloured path is rainbow if the colours of its edges are distinct. For a positive integer \( k \), an edge-colouring of a graph \( G \) is rainbow \( k \)-connected if any two vertices of \( G \) are connected by \( k \) internally vertex-disjoint rainbow paths. The rainbow \( k \)-connection number \( rc_k(G) \) is defined to be the minimum integer \( t \) such that there exists an edge-colouring of \( G \) with \( t \) colours which is rainbow \( k \)-connected. We consider \( rc_2(G) \) when \( G \) has fixed vertex-connectivity. We also consider \( rc_k(G) \) for large complete bipartite and multipartite graphs \( G \) with equipartitions. Finally, we determine sharp threshold functions for the properties \( rc_k(G) = 2 \) and \( rc_k(G) = 3 \), where \( G \) is a random graph. Related open problems are posed.