  Our first goal in this work is to study general and model-independent
properties of cyclic cosmologies. The large number of studies of bouncing
cosmologies and different cyclic scenarios published recently calls for a
proper understanding of the universal properties of cyclic models. We thus
first review and further elaborate the common physical and geometrical
properties of various classes of cyclic models and then discuss how cyclic
Universe can be treated as a dynamic system. We then discuss how two theorems
from dynamic systems analysis can be used to ensure the existence of cyclic
cosmological solutions under certain conditions on the field equations. After
this we proceed towards our second goal which is the application of the
obtained results to different frameworks of modified gravity theories: $f(R)$
gravity, dynamic dark energy and $f(T)$ gravity. We discuss the general
requirements for the existence of cyclic solutions in these theories and also
obtain various examples of cyclic cosmologies, while discussing their basic
properties.
