This paper is devoted to study the existence of singularities of series expansions in Laguerre polynomials of holomorphic functions on the boundaries of its natural region of convergence. An Equiconvergence theorem is proved by which the convergence problem of Laguerre series can be reduced to the simpler discussion of the convergence of associated with them trigonometric series. Based on this the existence of Laguerre series having no finite singular points is established. A theorem is also proved which makes us sure that as inthe case for power series, for the analytic continuation of functions defined by series in Laguerre polynomials it can be judged by the properties of the series coefficients. The final section of the paper contains some proposals as a result of applying the classical method of constructing non-continuable series by means of gaps.