If you have a running Julia distribution (Julia 0.4 or greater), you have everything you need to install POMDPs.jl. To install the package, simply run the following from the Julia REPL:

import Pkg
Pkg.add("POMDPs") # installs the POMDPs.jl package

Some auxiliary packages and older versions of solvers may be found in the JuliaPOMDP registry. To install this registry, run:

using Pkg; pkg"registry add"

Note: to use this registry, JuliaPro users must also run edit(normpath(Sys.BINDIR,"..","etc","julia","startup.jl")), comment out the line ENV["DISABLE_FALLBACK"] = "true", save the file, and restart JuliaPro as described in this issue.