The paper is concerned with the single server queues N/G/1 and GI/NI/1, respectively, in which the arrival process or the service process is a Neuts process, and derives the matrix-exponential forms of the solution of relevant nonlinear matrix equations for such queues. It generalises the matrix-exponential results of \textit{B. Sengupta} [Adv. Appl. Probab. 21, No.1, 159-180 (1989; Zbl 0672.60090)] for GI/PH/1 and of \textit{M. Neuts} [see ``Structured stochastic matrices of M/G/1 type and their applications.'' (1989; Zbl 0695.60088), and ``Matrix-geometric solutions in stochastic models. An algorithmic approach.'' (1981; Zbl 0469.60002)] for MMPP/G/1 to substantially more general models. The derivation of the results also establishes the equivalence of the methods of Neuts and those of Sengupta. A detailed analysis of the queue GI/N/1 is given, and it is noted that not only the stationary distribution at arrivals but also at an arbitrary time is matrix-geometric. Matrix-exponential steady state distributions are established for the waiting times in the queue GI/N/1. From this, by appealing to the author's duality theorem [Commun. Stat., Stochastic Models 6, No.1, 151-161 (1990; Zbl 0699.60091)] it is deduced that the stationary virtual and actual waiting times in a GI/PH/1 queue are of phase type.