This paper is a detailed report on a programme of direct numerical simulations of incompressible nonhelical randomly forced magnetohydrodynamic (MHD) turbulence that are used to settle a long-standing issue in the turbulent dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm 1 and small magnetic Prandtl number Pm 1. The dependence of the critical Rm_c for dynamo versus the hydrodynamic Reynolds number Re is obtained for 1 Re 6700. In the limit Pm 1, Rm_c is at most three times larger than for the previously well established dynamo at large and moderate Prandtl numbers: Rm_c 200 for Re 6000 compared to Rm_c ~ 60 for . The stability curve Rm_c(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate of the magnetic energy is in the limit Re Rm 1, as should be the case if the dynamo is driven by the inertial-range motions at the resistive scale, or tends to an Rm-independent value comparable to the turnover rate of the outer-scale motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm ≥ 1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At , the magnetic fluctuations induced via the tangling by turbulence of a weak mean field are investigated and the possibility of a k⁻¹ spectrum above the resistive scale is examined. At low Rm < 1, the induced fluctuations are well described by the quasistatic approximation; the k^−11/3 spectrum is confirmed for the first time in direct numerical simulations. Applications of the results on turbulent induction to understanding the nonlocal energy transfer from the dynamo-generated magnetic field to smaller-scale magnetic fluctuations are discussed. The results reported here are of fundamental importance for understanding the genesis of small-scale magnetic fields in cosmic plasmas.