  We prove the existence and uniqueness of harmonic maps in degree one homotopy
classes of closed, orientable surfaces of positive genus, when the target has
conic points with cone angles less than $2\pi$. For a cone point $p$ of cone
angle less than or equal $\pi$ we show that one can minimize, uniquely, in the
relative homotopy class of a homeomorphism sending a fixed point $q$ in the
domain to $p$. The latter can be interpreted as minimizing maps from punctured
Riemann surfaces.
