  In this thesis we describe the universal central extension of two important
classes of so-called root-graded Lie algebras defined over a commutative
associative unital ring $k.$ Root-graded Lie algebras are Lie algebras which
are graded by the root lattice of a locally finite root system and contain
enough $\mathfrak{sl}_2$-triples to separate the homogeneous spaces of the
grading. Examples include the infinite rank analogs of the simple
finite-dimensional complex Lie algebras. \\ In the thesis we show that in
general the universal central extension of a root-graded Lie algebra $L$ is not
root-graded anymore, but that we can measure quite easily how far it is away
from being so, using the notion of degenerate sums, introduced by van der
Kallen. We then concentrate on root-graded Lie algebras which are graded by the
root systems of type $A$ with rank at least 2 and of type $C$. For them one can
use the theory of Jordan algebras.
