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We report results from a multi-wavelength monitoring campaign on Mrk 421 overthe period of 2003-2004. The source was observed simultaneously at TeV andX-ray energies, with supporting observations frequently carried out at opticaland radio wavelengths. The large amount of simultaneous data has allowed us toexamine the variability of Mrk 421 in detail. The variabilities are generallycorrelated between the X-ray and gamma-ray bands, although the correlationappears to be fairly loose. The light curves show the presence of flares withvarying amplitudes on a wide range of timescales both at X-ray and TeVenergies. Of particular interest is the presence of TeV flares that have nocoincident counterparts at longer wavelengths, because the phenomenon seemsdifficult to understand in the context of the proposed emission models for TeVblazars. We have also found that the TeV flux reached its peak days before theX-ray flux during a giant flare in 2004. Such a difference in the developmentof the flare presents a further challenge to the emission models. Mrk 421varied much less at optical and radio wavelengths. Surprisingly, the normalizedvariability amplitude in optical seems to be comparable to that in radio,perhaps suggesting the presence of different populations of emitting electronsin the jet. The spectral energy distribution (SED) of Mrk 421 is seen to varywith flux, with the two characteristic peaks moving toward higher energies athigher fluxes. We have failed to fit the measured SEDs with a one-zone SSCmodel; introducing additional zones greatly improves the fits. We have derivedconstraints on the physical properties of the X-ray/gamma-ray flaring regionsfrom the observed variability (and SED) of the source. The implications of theresults are discussed. (Abridged)Comment: 32 pages, 12 figures, to appear in Ap

A Multiwavelength View of the TeV Blazar Markarian 421: Correlated Variability, Flaring, and Spectral Evolution