  In this paper we study quasi-homogeneous operators, which include the
homogeneous operators, in the Cowen-Douglas class. We give two separate
theorems describing canonical models (with respect to equivalence under unitary
and invertible operators, respectively) for these operators using techniques
from complex geometry. This considerably extends the similarity and unitary
classification of homogeneous operators in the Cowen-Douglas class obtained
recently by the last author and A. Kor\'{a}nyi. Specifically, the complex
geometric invariants used for our classification are the curvature and the
second fundamental forms inherent in the definition of a quasi-homogeneous
operator. We show that these operators are irreducible and determine when they
are strongly irreducible. Applications include the equality of the topological
and algebraic K-group of a quasi-homogeneous operator and an affirmative answer
to a well-known question of Halmos on similarity for these operators.
