We present an overview of some results we have obtained recently (A. Sain, Manu and R. Pandit, Phys. Rev. Lett. 81 (1998) 4377) from a pseudospectral study of the randomly forced Navier-Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance similar to k(4-d-y). With k the wavevector and the dimension d = 3, These include the multiscaling of velocity structure functions for y greater than or equal to 4 and a demonstration that the multiscaling exponent ratios zeta(p)/zeta(2) for y = 4 are in agreement, with those obtained for the Navier-Stokes equation forced at large spatial scales (3dNSE). We also study a coarse-graining procedure for the 3aNSE and examine why it does not lead to the RFNSE. (C) 1999 Elsevier Science B.V. All rights reserved.