We analyze star formation (SF) as a function of stellar mass (M*) and redshift z in the All Wavelength Extended Groth Strip International Survey (AEGIS). For 2905 field galaxies, complete to 10^10(10^10.8) Msun at z<0.7(1), with Keck spectroscopic redshifts out to z=1.1, we compile SF rates (SFR) from emission lines, GALEX, and Spitzer MIPS 24 micron photometry, optical-NIR M* measurements, and HST morphologies. Galaxies with reliable signs of SF form a distinct main sequence (MS), with a limited range of SFR at a given M* and z (1 sigma < +-0.3 dex), and log(SFR) approximately proportional to log(M*). The range of log(SFR) remains constant to z>1, while the MS as a whole moves to higher SFR as z increases. The range of SFR along the MS constrains the amplitude of episodic variations of SF, and the effect of mergers on SFR. Typical galaxies spend ~67(95)% of their lifetime since z=1 within a factor of <~ 2(4) of their average SFR at a given M* and z. The dominant mode of the evolution of SF since z~1 is apparently a gradual decline of the average SFR in most individual galaxies, not a decreasing frequency of starburst episodes, or a decreasing factor by which SFR are enhanced in starbursts. LIRGs at z~1 seem to mostly reflect the high SFR typical for massive galaxies at that epoch. The smooth MS may reflect that the same set of few physical processes governs star formation prior to additional quenching processes. A gradual process like gas exhaustion may play a dominant role.
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