This correspondence deals with self diagnosis and system level diagnosis of regular arrays of N processor units, where each unit can test its L neighbors. It is first shown that such an array is one-step L-fault diagnosable iff N 2L + 1. To produce an array which is t-fault diagnosable, for larger values of t than L, the following four methods are investigated: (1) Limit the fault patterns in the system; (2) Employ sequential diagnosis; (3) Employ inexact (t/s) diagnosis; (4) Make the diagnosis independent of t. If the distance between faulty units is at least d + 1, then an n × n array is one-step n n/(d + 1) — fault diagnosable if d > 1 and n > d + 1. Simple algorithms for identifying faulty units are presented. If N > t, then the array is at most t/L — step t-fault diagnosable. Also, the array is one step t/s diagnosable iff N s + L + 1. The exact expression for s is presented, and in most cases s ≈ t²/4L. A relationship between diagnosis resolution, i.e. the number of good units diagnosed as potentially being faulty, and the number of steps required in the sequential diagnostic procedure is derived. By increasing the number of steps in the test process, the diagnostic resolution can be reduced. The paper concludes with some comments on the control unit required to decode the syndrome and identify good and faulty units.