  This paper studies the magnitude homology groups of geodesic metric spaces.
We start with a description of the second magnitude homology of a general
metric space in terms of the zeroth homology groups of certain simplicial
complexes. Then, on a geodesic metric space, we interpret the description by
means of geodesics. The third magnitude homology of a geodesic metric space
also admits a description in terms of a simplicial complex. Under an assumption
on a metric space, the simplicial description allows us to introduce an
invariant of third magnitude homology classes as an intersection number.
Finally, we provide a complete description of all the magnitude homology groups
of a geodesic metric space which fulfils a certain non-branching assumption.
