Several natural constructions of holomorphic mappings of Riemann surfaces into loop spaces endowed with natural almost complex structures will be presented. The main attention will be given to based loop spaces of compact Lie groups and immersed loop spaces of three-dimensional Riemannian manifolds. In particular case when the source space is the unit disc in complex plane, basic geometric properties of the arising holomorphic families of loops will be established and their interpretation in dynamical terms will be given. It will be shown that examples of holomorphic dynamics in loop spaces arise from Seifert fibrations and isolated singularities of algebraic plane curves. A visual example provided by the inverse of Hopf fibration will be considered in some detail. It will be also shown that each Seifert fibration can be turned into a holomorphic family of loops parameterized by its factor-space.