In this paper, online convex optimization problem with time-varying constraints is studied from the perspective of an agent taking sequential actions. Both the objective function and the constraint functions are dynamic and unknown a priori to the agent. We first consider the scenario of the gradient feedback, in which, the values and gradients of the objective function and constraint functions at the chosen action are revealed after an action is submitted. We propose a computationally efficient online algorithm, which only involves direct closed-form computations at each time instant. It is shown that the algorithm possesses sublinear regret with respect to the dynamic benchmark sequence and sublinear constraint violations, as long as the drift of the benchmark sequence is sublinear, or in other words, the underlying dynamic optimization problems do not vary too drastically. Furthermore, we investigate the scenario of the bandit feedback, in which, after an action is chosen, only the values of the objective function and the constraint functions at several random points close to the action are announced to the agent. A bandit version of the online algorithm is proposed and we also establish its sublinear expected regret and sublinear expected constraint violations under the assumption that the drift of the benchmark sequence is sublinear. Finally, two numerical examples, namely online quadratic programming and online logistic regression, are presented to corroborate the effectiveness of the proposed algorithms and to confirm the theoretical guarantees.