  We compute explicitly the group of connected components $\pi_0G(\mathbb{R})$
of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear)
connected algebraic group $G$ defined over the field $\mathbb{R}$ of real
numbers. In particular, it turns out that $\pi_0G(\mathbb{R})$ is always an
elementary Abelian 2-group. The result looks most transparent in the cases
where $G$ is a linear algebraic group or an Abelian variety. The computation is
based on structure results on algebraic groups and Galois cohomology methods.
