  The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the
second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler
value among all planar graphs $G$ with $n$ vertices, denoted by
$\lambda_{2\max}$, and we show the bounds $2+\Theta(\frac{1}{n^2}) \leq
\lambda_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum
Fiedler value for the following classes of planar graphs: Bipartite planar
graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar
graphs. Furthermore, we derive almost tight bounds on $\lambda_{2\max}$ for two
more classes of graphs, those of bounded genus and $K_h$-minor-free graphs.
