Title:  Colourful linear programming and simplicial depth

We present recent results and open questions dealing with a generalization of linear programming introduced by Bárány and Onn in 1997, and the associated generalization of the Carathéodory's Theorem proven by Bárány in 1982. In particular, we present recent generalizations of the Colourful Carathéodory Theorem due to Arocha et al. and Holmsen et al. and our strengthening of these. We discuss a combinatorial approach that led to the recent determination by Sarrabezolles of a tight lower bound for the colourful simplicial depth in any dimension and generalizing earlier results due to Bárány and Matoušek, and Stephen and Thomas. Based on joint works with Frédéric Meunier (ENPC Paris) and Pauline Sarrabezolles (ENPC Paris).