One fundamental issue in sensor networks is the coverage problem, which reflects how well a sensor network is monitored or tracked by sensors. In this paper, we formulate this problem as a decision problem, whose goal is to determine whether every point in the service area of the sensor network is covered by at least k sensors, where k is a predefined value. The sensing ranges of sensors can be unit disks or non-unit disks. We present polynomial-time algorithms, in terms of the number of sensors, that can be easily translated to distributed protocols. The result is a generalization of some earlier results where only k=1 is assumed. Applications of the result include: (i) positioning applications, (ii) situations which require stronger environmental monitoring capability, and (iii) scenarios which impose more stringent fault-tolerant capability.