Let (W; L, L') be a monotone Lagrangian cobordism. In some cases, for example when the minimal Maslov number of W is big enough, the fundamental groups of L and L' give a lot of information about the topology of $W$. Motivated by this question, we study the fundamental group of W using two approaches: a geometric approach based on Barraud's construction of the Floer fundamental group and an algebraic approach using Floer homology with local coefficients. This talk is based on a joint work with Jean-François Barraud.