  We investigate the curvature of Eschenburg spaces with respect to two
different metrics, one constructed by Eschenburg and the other by Wilking. We
show that, with precisely one exception, that Wilking's metric on an Eschenburg
space of cohomogeneity at most two is almost positively curved. This provides
the first infinite family of almost positively curved examples since Wilking
first introduced his construction. In addition, we completely characterize the
curvature of every Eschenburg space with respect to Eschenburg's metric.
