This paper presents a general theory for the fields generated by a circular current loop and compares it with existing theories. The existing, general, closed solution for the vector magnetic field may be expressed in a number of seemingly different but equivalent forms. These relations offer alternative closed-form solutions that may find various applications, including the characterization of Helmholtz coils. The paper provides alternative closed forms in both spherical and cylindrical coordinate systems. It employs Gauss's magnetic law to show that the alternative closed form is self-consistent and correct and also shows agreement with well-known solutions. Finally, it develops a new (or not readily found) tabulated mathematical identity.