In this dissertation we treat some problems about possible density of orbits for non-hypercyclic operators and we enlarge the class of known non-orbit-transitive operators. One of the questions related to hypercyclic operators that we answer is whether the density (in the set of positive real numbers) of the norms of the elements in the orbit for each nonzero vector in the Hilbert space is sufficient to imply that at least one vector has orbit dense in the Hilbert space. We show that the density of the norms is not a sufficient condition to imply hypercyclicity by constructing a weighted bilateral shift that, on one hand, satisfies the orbit-density property (in the sense defined above), but, on the other hand, fails to be hypercyclic. The second major topic that we study refers to classes of operators that are not hypertransitive (or orbit-transitive) and is related to the invariant subspace problem on Hilbert space. It was shown by Jung, Ko and Pearcy in 2005 that every compact perturbation of a normal operator is not hypertransitive. We extend this result, after introducing the related notion of weak hypertransitivity, by giving a sufficient condition for an operator to belong to the class of non-weakly-hypertransitive operators. Next, we study certain 2-normal operators and their compact perturbations. In particular, we consider operators with a slow growth rate for the essential norms of their powers. Using a new idea, of accumulation of growth for each given power on a set of different orthonormal vectors, we establish that the studied operators are not hypertransitive.