Abstract: In this talk we will introduce some fundamental results in semigroup theory (spectrum, spectral bound, growth bound, resolvent, Laurent’s expansion, integral resolvent formula) and spectral theory for linear operators. Then we will present some basics on abstract semilinear equations using semigroups and spectral theory and show that several types of differential equations, such as functional differential equation, age-structured models, transport equations, reaction-diffusion equations, and partial differential equations with delay, can be formulated as abstract Cauchy problems with non-dense domain.