  Catalytic coherence transformations allow the otherwise impossible state
transformations using only incoherent operations with the aid of an auxiliary
system with finite coherence which is not being consumed in anyway. Here we
find the necessary and sufficient conditions for the deterministic and
stochastic catalytic coherence transformations between pair of pure quantum
states. In particular, we show that the simultaneous decrease of a family of
R\'enyi entropies of the diagonal parts of the states under consideration are
necessary and sufficient conditions for the deterministic catalytic coherence
transformations. Similarly, for stochastic catalytic coherence transformations
we find the necessary and sufficient conditions for achieving higher optimal
probability of conversion. We, thus, completely characterize the coherence
transformations amongst pure quantum states under incoherent operations. We
give numerous examples to elaborate our results. We also explore the
possibility of the same system acting as a catalyst for itself and find that
indeed {\it self catalysis} is possible. Further, for the cases where no
catalytic coherence transformation is possible we provide entanglement assisted
coherence transformations and find the necessary and sufficient conditions for
such transformations.
