In this paper I studied Fredholm alternative and its consequences. The Fredholm alternative studies the Fredholm integral equation together with the associated transposed equation, and the associated homogeneous and homogeneous transposed equation. The conditions for the existence of a solution by an alternative are divided into two parts, depending on the fulfilment of certain conditions imposed by the alternative itself as a theorem. Integral equations with degenerate kernels, i.e. kernels that can be represented as the sum products of functions of only \(x\) and only \(t\) variables lead to a system of linear equations, which are then studied by standard methods, i.e. the rank and defect theorem is used. A case with more general kernel, i.e. not necessarily degenerate, can be reduced to a case with degenerate kernel if we approximate the arbitrary non-degenerate kernel with a degenerate one. Using methods of modern functional analysis it is possible to study Fredholm integral equations with integral operators and we list some properties of these operators. With the introduction of integral operators, the problem of finding the solution of the integral equation leads to the well-known problem of finding the eigenvalues and eigenvectors of an operator.