  Functional effects of different mutations are known to combine to the total
effect in highly nontrivial ways. For the trait under evolutionary selection
(`fitness'), measured values over all possible combinations of a set of
mutations yield a fitness landscape that determines which mutational states can
be reached from a given initial genotype. Understanding the accessibility
properties of fitness landscapes is conceptually important in answering
questions about the predictability and repeatability of evolutionary
adaptation. Here we theoretically investigate accessibility of the globally
optimal state on a wide variety of model landscapes, including landscapes with
tunable ruggedness as well as neutral `holey' landscapes. We define a
mutational pathway to be accessible if it contains the minimal number of
mutations required to reach the target genotype, and if fitness increases in
each mutational step. Under this definition accessibility is high, in the sense
that at least one accessible pathwayexists with a substantial probability that
approaches unity as the dimensionality of the fitness landscape (set by the
number of mutational loci) becomes large. At the same time the number of
alternative accessible pathways grows without bound. We test the model
predictions against an empirical 8-locus fitness landscape obtained for the
filamentous fungus \textit{Aspergillus niger}. By analyzing subgraphs of the
full landscape containing different subsets of mutations, we are able to probe
the mutational distance scale in the empirical data. The predicted effect of
high accessibility is supported by the empirical data and very robust, which we
argue to reflect the generic topology of sequence spaces.
