Characterize where the Dirichlet Problem for the Laplacian is always solvable Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner characterization is known.  Has any progress been made since then? And what simpler characterization is known today, if one is known?
Here is the problem definition:
An open connected set $G\subseteq \mathbb{C}$ is called a Dirichlet Region if for each continuous function $f:\partial_\infty G\rightarrow \mathbb{R}$ there is a continuous function $u:G^- \rightarrow \mathbb{R}$ such that $u$ is harmonic in $G$ and $u(z)=f(z)$ for all $z$ in $\partial_\infty G$.
(The notation $\partial_\infty G$ refers to the boundary of $G$ in $\mathbb{C}\cup\{\infty\}$, and $G^-$ denotes the closure of $G$ in $\mathbb{C}\cup\{\infty\}$.)
The characterization given in the book is:
Given $a \in \partial_\infty G$, a barrier for $G$ at $a$ is a family $\{\psi_r: r>0\}$ of functions such that: 
1. $\psi_r$ is well-defined and superharmonic on $B(a;r) \cap G$ with $0\leq \psi_r(z) \leq 1$ 
2. $\lim_{z\rightarrow a}\psi_r(z) = 0$, and 
3. $\lim_{z\rightarrow w} \psi_r(z) = 1$ for $w$ in $G \cap \{w:|w-a|=r\}$.
An open connected set $G$ is a Dirichlet Region iff there is a barrier for $G$ at each point of $\partial_\infty G$.  
 A: I assume you are asking about strong solutions (so u is actually $C^2(G)\cap C(\partial_\infty G))$. In this case, the characterization via barriers, or equivalently, as Will says,  using  Perron's method, cannot be improved upon, I think. 
Here's what I remember, please correct me if there are flaws in the argument.
Define a regular point as a point $a$ on the boundary of $G$ such that a barrier exists at $a$ (with respect to G).Conway's characterization is saying domains with boundaries consisting of only regular points are Dirichlet regions. 
Now for the converse: if a region is a Dirichlet region, it must have a boundary of regular points. Suppose there is a domain $G$ which is a Dirichlet region. Let  $y$ be a point on the boundary of the domain. Consider a continuous function $f:\partial_\infty G \rightarrow \mathbb{R}$ such that $f(y)=0$, and $f>0$ for all other parts of the boundary.The solution $u$ of the Dirichlet problem with $f$ as data is, by the strong maximum principle, a barrier at $y$. Hence $y$ is a regular point.
The question of the boundary regularity necessary for solvability of the Dirichlet problem has indeed been studied, and the answer may vary depending on the specific notion of solvability (strong solution? weak solution? solution a.s.?). Gilbarg and Trudinger, H\"ormander, Maz'ya have all written nice books on this and related topics. 
A: http://eom.springer.de/r/r080680.htm contains some characterizations of domains where the Dirichlet problem is solvable. I believe that a key term in searching for references is "Wiener's criterion."
