  We determine the fundamental groups of symmetrizable algebraically simply
connected split real Kac-Moody groups endowed with the Kac-Peterson topology.
In analogy to the finite-dimensional situation, the Iwasawa decomposition $G =
KAU_+$ provides a weak homotopy equivalence $K \hookrightarrow G$, implying
$\pi_1(G) = \pi_1(K)$. It thus suffices to determine $\pi_1(K)$ which we
achieve by investigating the fundamental groups of generalized flag varieties.
Our results apply in all cases in which the Bruhat decomposition of the
generalized flag variety is a CW decomposition $-$ in particular, we cover the
complete symmetrizable situation; the result concerning the structure of
$\pi_1(K)$ more generally also holds in the non-symmetrizable two-spherical
situation.
