  We use a sample of rest-frame UV selected and spectroscopically observed
galaxies at redshifts 1.9<z<3.4, combined with ground-based spectroscopic
H-alpha and Spitzer MIPS 24 micron data, to derive the most robust measurements
of the rest-frame UV, H-alpha, and infrared (IR) luminosity functions (LFs) at
these redshifts. Our sample is by far the largest of its kind, with over 2000
spectroscopic redshifts in the range 1.9<z<3.4 and ~15000 photometric
candidates in 29 independent fields covering a total area of almost a square
degree. Our method for computing the LFs takes into account a number of
systematic effects, including photometric scatter, Ly-alpha perturbations to
optical colors, and contaminants. Taking into account the latter, we find no
evidence for an excess of UV-bright galaxies over what was inferred in early
z~3 LBG studies. The UV LF appears to undergo little evolution between z~4 and
z~2. Corrected for extinction, the UV luminosity density (LD) at z~2 is at
least as large as the value at z~3 and a factor of ~9 larger than the value at
z~6, primarily reflecting an increase in the number density of bright galaxies
between z~6 and z~2. Our analysis yields the first constraints anchored by
extensive spectroscopy on the IR and bolometric LFs for faint and moderately
luminous (L[bol]<10^12 L_sun) galaxies. Adding the IR to the emergent UV
luminosity, incorporating independent measurements of the LD from ULIRGs, and
assuming realistic dust attenuation values for UV-faint galaxies, indicates
that galaxies with L[bol]<10^12 L_sun account for ~80% of the bolometric LD and
SFRD at z~2-3. Our multi-wavelength constraints on the global SFRD indicate
that approximately one-third of the present-day stellar mass density was formed
in sub-ultraluminous galaxies between redshifts z=1.9-3.4. [Abridged]
