Theory and application of stochastic approximation (SA) have become increasingly relevant due in part to applications in optimization and reinforcement learning. This paper takes a new look at SA with constant step-size $α>0$, defined by the recursion, $$θ_{n+1} = θ_{n}+ αf(θ_n,Φ_{n+1})$$ in which $θ_n\in\mathbb{R}^d$ and $\{Φ_{n}\}$ is a Markov chain. The goal is to approximately solve root finding problem $\bar{f}(θ^*)=0$, where $\bar{f}(θ)=\mathbb{E}[f(θ,Φ)]$ and $Φ$ has the steady-state distribution of $\{Φ_{n}\}$. The following conclusions are obtained under an ergodicity assumption on the Markov chain, compatible assumptions on $f$, and for $α>0$ sufficiently small: $\textbf{1.}$ The pair process $\{(θ_n,Φ_n)\}$ is geometrically ergodic in a topological sense. $\textbf{2.}$ For every $1\le p\le 4$, there is a constant $b_p$ such that $\limsup_{n\to\infty}\mathbb{E}[\|θ_n-θ^*\|^p]\le b_p α^{p/2}$ for each initial condition. $\textbf{3.}$ The Polyak-Ruppert-style averaged estimates $θ^{\text{PR}}_n=n^{-1}\sum_{k=1}^{n}θ_k$ converge to a limit $θ^{\text{PR}}_\infty$ almost surely and in mean square, which satisfies $θ^{\text{PR}}_\infty=θ^*+α\barΥ^*+O(α^2)$ for an identified non-random $\barΥ^*\in\mathbb{R}^d$. Moreover, the covariance is approximately optimal: The limiting covariance matrix of $θ^{\text {PR}}_n$ is approximately minimal in a matricial sense. The two main take-aways for practitioners are application-dependent. It is argued that, in applications to optimization, constant gain algorithms may be preferable even when the objective has multiple local minima; while a vanishing gain algorithm is preferable in applications to reinforcement learning due to the presence of bias.