  In this article we shall survey some recent progress on the study of
Ap\'ery-like sums which are multiple variable generalizations of the two sums
Ap\'ery used in his famous proof of the irrationality of $\zeta(2)$ and
$\zeta(3)$. We only allow the central binomial coefficients to appear in these
infinite sums but they can appear either on the numerator or on the
denominator. Special values of both types are closely related to the colored
multiple zeta values and have played important roles in the calculations of the
$\eps$-expansion of multiloop Feynman diagrams. We will summarize several
different approaches to computing these sums and prove a few conjectural
identities of Z.-W. Sun as corollaries along the way.
