Preliminary Concepts. Stieltjes Integrals and Fourier Series: 1 Functions of limited variation; 2 Continuation of the preceding; 3 Integrals with respect to a function of limited variation; 4 Note on the second law of the mean; 5 Classical theorems on integrals and limits of integrals; 6 The limits of Stieltjes integrals; 7 Note on Lebesgue integrals; 8 Convergence of Fourier series; 9 Summability of series Functions Harmonic Within a Circle: 10 Preliminary theorems; 11 A preliminary result; 12 Note on integral identities; 13 Digression: Functions of points and of point sets; 14 Properties of the Poisson-Stieltjes integral; 15 Continuation of the preceding: Behaviour of $u(r, \theta)$ in the neighborhood of the boundary; 16 The Poisson integral: $F(\varphi)$ absolutely continuous Necessary and Sufficient Conditions. The Dirichlet Problems for the Circle: 17 Fundamental theorem and lemma; 18 Proof of fundamental theorem; 19 Special cases of the Poisson integral; 20 The Dirichlet problem and its extension Potentials of a Single Layer and the Neumann Problem: 21 The Stieltjes integral for potentials of a single layer; 22 Necessary and sufficient conditions; 23 Further properties; 24 The Neumann problems; 25 General points of view; 26 Digression: Physical interpretation of a general distribution of mass; 27 Cauchy's integral formula General Simply Connected Plane Regions and the Order of Their Boundary Points: 28 Conformal transformations and general regions; 29 Invariant forms of conditions (i), (ii) etc.; 30 Invariant forms of conclusions; 31 Order of boundary points; 32 Integrals on the boundary and the Dirichlet problems; 33 Special cases of the condition (ii). The continuous boundary value problem; 34 A new continuous boundary value problem; 35 The generalized Neumann problem in the general region Plane Regions of Finite Connectivity: 36 Functions harmonic outside a circle; 37 The multiply connected region bounded by $n+1$ distinct circles; 38 Representation in terms of the Green's function; 39 Boundary integrals and Stieltjes integral equations; 40 General regions of finite connectivity. Isolated point boundaries; 41 Annular regions. Determination of the functions $F 0(\theta)$ and $F 1(\theta)$; 42 Uniqueness of the representation of Theorem 3 for $S$ Related Problems: 43 A simple discontinuous boundary value problem; 44 Continuous boundary value problems; 45 Regions with continuous boundaries; 46 Regions with rectifiable boundaries; 47 Regions of infinite connectivity; 48 Remarks on necessary and sufficient conditions; 49 Convergence in the mean of positive order less than one; 50 Integro-differential equations of Bocher type.