Biweekly timing observations of PSRs B1855+09 and B1937+21 have been made at the Arecibo Observatory for more than 7 and 8 year, respectively, with uniform procedures and only a few modest gaps. On each observing date we measure an equivalent pulse arrival time for PSR B1855+09 at 1.4 GHz, with typical accuracies of about 0.8 micrometers and for PSR B1937 + 21 at both 1.4 and 2.4 GHz, with accuracies around 0.2 micrometers. The pulse arrival times are fitted to a simple model for each pulsar, yielding high-precision astrometric, rotational, and orbital parameters, and a diverse range of conclusions. The celestial coordinates and proper motions of the two pulsars are determined with uncertainties less than or equal to 0.12 mas and less than or equal to 0.06 mas/year in the reference frame of the DE200 planetary ephemeris. The annual parallaxes are found to be pi=1.1 +/- 0.3 mas and pi less than 0.28 mas for PSRs B1855+09 and B1937+21, respectively. The general relativistic Shapiro delay is measured in the PSR B1855+09 system and used to obtain masses m₁ = 1.50 ^+0.26 _-0.14 solar mass and m (sub2) = 0.258 ^+0.028_-0.016solar mass for the pulsar and its orbiting companion. The extremely stable orbital period of this system provides a phenomenological limit on the secular change of Newton's gravitational constant, dot-G/G=(-9 +/- 18) x 10^-12/year. Variations in the dispersion measure of PSR B1937 + 21 indicate that the spectrum of electron-density fluctuations in the interstellar medi um has a power-law index beta = 3.874 +/- 0.011, slightly steeper than the Kolmogorov value of 11/3, and we find no strong evidence for an 'inner scale' greater than about 2 x 10⁹cm. In the residual pulse arrival times for PSR B1937+21 we have observed small systematic trends not explained by our deterministic timing model. We discuss a number of possible causes; although the results are not yet conclusive, the most straightforward interpretation is that the unmodeled noise (a few microseconds over 8 yr, or Delta t/T approximately equals 10^-14 is inherent to the pulsar itself. In the present data set, PSR B1855+09 exhibits no discernible timing noise. With conventional assumptions we derive a limit Omega _g h² less than 6 x 10^-8 (95% confidence) for the energy density, per logarithmic frequency interval, in a cosmic background of stochastic gravitational waves. We discuss the feasibility of establishing a pulsar-based timescale that might be used to test the stabilities of the best available atomic clocks. In an Appendix, we propose guidelines for the archiving of pulsar timing observations. Instructions are provided for obtaining copies of our own archival data, via Internet.

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