Let $M$ be a compact Riemannian manifold with a metric $g$. Then there are natural Laplace-type operators associated to $g$ such as the de-Rham Laplacian on forms and the conformal Laplacian (Yamabe operator). For such an operator one can form the zeta function $Z(s)=\sum_j \la_j^{-s}$ where $\la_1,\,\la_2,\,\dots$ are the eigenvalues. For real $s$ fixed, the value $Z(s)$ or more generally $d^kZ/ds^k (s)$ is a real valued functional on the space of metrics. For a general zeta invariant, we show how to determine the type of the critical metrics up to finite index. A number of familiar functionals from Riemannian geometry are zeta invariants and we discuss some past, recent and potential applications of zeta invariants to geometry.