The variable-star population of the Galactic globular cluster NGC 1851 (C0512-400) has been studied by CCD photometry, from observations made in the B, V, and I bands during 1993-1994. Light curves are presented for 29 variables, seven of which are new discoveries. The behavior of the RR Lyrae stars in the period-temperature diagram appears normal when compared with clusters that bracket the NGC 1851 metallicity. Reddening and metallicity are reevaluated, with no compelling evidence being found to change from the values of E(B-V) = 0.02 and [Fe/H] = -1.29 (Zinn scale) adopted in recent studies of the cluster. Photometry is provided for stars in an annulus with radii 80 " and 260 " centered on NGC 1851. To at least V = 18.5 there is excellent agreement with the extensive earlier photometry for the brighter NGC 1851 stars, with systematics less than 0.02 mag in all colors. Instability-strip boundary positions for several clusters show a trend for the red boundary to move to redder colors as the metallicity increases.