Let F be a natural exponential family (NEF) generated by a measure μ and X = ( X 1 , … , X n ) a random sample with a common distribution belonging to F . Consider the set of order statistics X ( 1 ) ≤ X ( 2 ) ≤ ⋯ ≤ X ( n ) and let G r , n denote the family of distributions induced by the r -th order statistic X ( r ) , r = 1 , … , n . The main problem of the paper, namely, the closedness of NEF’s under the formation of order statistics, can be posed as follows: for which NEF’s F , the set of distributions G r , n constitutes, for all n ∈ N and for some r ∈ { 1 , … , n } , an NEF on R ? If G r , n is an NEF, we shall say that F is closed under the r -th order statistic. A comprehensive answer to this problem seems to be rather difficult when μ is an arbitrary measure. However, if μ is a continuous measure we show that if 1 < r < n , then G r , n is not an NEF. The remaining cases r = 1 or r = n are equivalent under an appropriate affine transformation. For r = n we prove that G n , n is an NEF if and only if F is the family of exponential distributions supported on R − ; or, equivalently, for r = 1 , G 1 , n is an NEF if and only if F is the family of exponential distributions supported on R + .