The aim of this dissertation is to construct new classes of infinite dimensional Lie groups and to show their regularity in Milnor's sense. This will imply the regularity of some Lie groups which where already constructed but not known to be regular so far. In Chapter one we fix some notation and give some basic definitions. In Chapter two we prove our main tool for both constructing new Lie groups and showing regularity of them. Theorem 2.1 is a sufficient criterion for mappings defined on direct limits of normed spaces to be complex analytic. In 1976, Pisanelli showed that the germs of holomorphic diffeomorphisms in C^n form an infinite dimensional Lie group.In Chapter three, we will generalize this to germs of analytic diffeomorphisms around a compact set in a Banach space. Furthermore, we will show that all Lie groups obtained in this fashion are regular. This implies regularity of Pisanelli's original example, which has been an open problem before. A result by Glöckner shows that the directed union of a sequence of finite dimensional Lie groups can always be given an (LB)-Lie group structure. In Chapter four, we show how to construct regular Lie group structures on ascending unions of a sequence of Banach Lie groups.In Chapter five, we will give some examples of cases where the situation of chapter four occurs.