This chapter focuses on the Laplace equation, \(\nabla^{2} V = 0\) , a fundamental equation in electrostatics and electrodynamics. Solutions in spherical coordinates are explored using the separation of variable technique, dividing the problem into radial, polar and azimuthal components. Key concepts include Legendre polynomials, derived from the Legendre differential equation and Rodrigue’s formula, which naturally emerge in systems with spherical symmetry. Boundary conditions are analysed in both spherical and cylindrical coordinate systems. In spherical symmetry, solutions address potentials inside or outside spherical conductors, while cylindrical symmetry involves cylindrical geometries and often employs Bessel functions. Practical applications demonstrate the Laplace equation’s utility in determining electrostatic potentials under specified boundary values or symmetries, such as potentials on spherical shells or cylindrical conductors. The chapter also covers generating functions and recursion relations for Legendre polynomials, which are critical in solving boundary value problems with spherical symmetry. The First and Second Uniqueness Theorems are demonstrated, ensuring the uniqueness of solutions to the Laplace equation under given boundary or charge distribution conditions, reinforcing its strength in physical applications.