Unconventional superconductivity and antiferromagnetism are often found in close proximity to one another. For the series of compounds Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, which possesses both antiferromagnetism and superconductivity for the under-doped range of composition, this observation is certainly true. The close proximity, and in fact coexistence for under-doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$, of antiferromagnetism and superconductivity has encouraged speculation that antiferromagnetic spin fluctuations may mediate the electron pairing interaction in unconventional superconductors. Previous studies indicated that the spin fluctuations at optimally-doped Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ are diffusive, while those at \bafeas are well defined spin wave excitations. Therefore, the nature of magnetic excitations in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ must change with the introduction of cobalt; but it is unclear if that change is merely a consequence of the loss of antiferromagnetic order, or a necessary ingredient for the appearance of superconductivity. To resolve this uncertainty, this work has been undertaken to study the spin fluctuations of five Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ compositions varying in cobalt concentration from lightly-doped to nearly optimally-doped as well as representative samples of other, non-superconducting, transition metal substituted \bafeas compounds. The spin fluctuations of these samples, in their antiferromagnetically ordered and (where possible) superconducting states have been studied via triple-axis and time-of-flight inelastic neutron scattering, and definitively determine the importance of spin fluctuations for superconductivity in Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$.