  In this paper we extend the concept of a conjugate point in a Riemannian
manifold to complete length spaces (also known as geodesic spaces). In
particular, we introduce symmetric conjugate points and ultimate conjugate
points. We then generalize the long homotopy lemma of Klingenberg to this
setting as well as the injectivity radius estimate also due to Klingenberg
which was used to produce closed geodesics or conjugate points on Riemannian
manifolds. Our versions apply in this more general setting. We next focus on
${\rm CBA}(\kappa)$ spaces, proving Rauch-type comparison theorems. In
particular, much like the Riemannian setting, we prove an Alexander-Bishop
theorem stating that there are no ultimate conjugate points less than $\pi$
apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison
theorem to precisely estimate the distance between nearby geodesics. We close
with applications and open problems.
