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We present a general theory for the origin and collimation of centrifugally driven, hydromagnetic winds from the surfaces of Keplerian accretion disks that surround central objects such as young stellar objects (YSOs) or black holes. The theory is fully two-dimensional and makes no assumption about self-similarity of the flows. These winds efficiently extract disk angular momentum and convert the gravitational binding energy released in the accretion process into the mechanical energy of the wind. We investigate solutions under the very general assumption that the magnetic flux in the disk scales as a power law in disk radius r_0_ as ψ_0_ is proportional to r^3/2α^. We find analytical solutions for many of the physical quantities in the outflow, such as the wind speed, mass-loss rate, thrust, density, Alfven surface, etc. We demonstrate that the self-similar solutions of Blandford & Payne (α = 2) are a special case of our theory. We find that large classes of models have the generic property that they self-collimate due to the J X B pinching force arising from the toroidal magnetic field component in the outflow that dominates far from the surface of the accretion disk. We also find that flows that achieve modest values for fast magnetosonic Mach numbers recollimate toward the axis forming magnetic focal points. We identify a unique non-self-similar solution (α = 3) that carries a constant current intensity everywhere (the toroidal field scales as B<SUB>phi</SUB> is proportional to r^-1^ for this case) and is free of the divergences of self-similar models. We apply our results to both bipolar outflows in the vicinity of YSOs and black hole/accretion disk models for active galactic nuclei (AGNs). We find that modest magnetic field strengths (<= 10^4^ G for AGNs, 10 G for YSOs) in the inner portions of these disks are sufficient to drive flows that match all the known observational constraints. In no case do the required magnetic energy densities exceed that of thermal pressure in these disks. An important new result in our theory is an intrinsic scale length to the disk outflow. The mass accretion rate is fundamentally connected to the wind mass-loss rate from the disk because it is the wind that carries off the disk angular momentum. We typically find that the ratio of the wind massloss rate to the accretion rate, M_w_/M_a_ is 10^-1^ for both the YSO and AGN models. Our results suggest that the highly collimated outflows seen in regions of star formation and in galactic nuclei share a simple physical mechanism that depends only upon the presence of magnetized gas with angular momentum, and the depth of the central gravitational potential well.