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In this first of two papers (hereinafter called Paper I), we present a universal approach for simply realizing monolithic photonic integration based on asymmetric twin-waveguide (ATG) technology. The concepts and important developments leading to ATG integration technology will be reviewed. The ATG structure consists of active and/or passive devices formed in separate, vertically displaced waveguides. Light is transferred between the waveguides via very low loss, lateral, adiabatic tapered mode transformers, allowing different optical functions to be realized in the different waveguides. The design of the adiabatic tapered mode transformer uses an algorithm based on perturbation theory. We show that the same designs can also be deduced from coupled local mode theory. Using the perturbation algorithm to design the taper coupler in an ATG based high bandwidth photodiode, a transfer efficiency of greater than 90% from the fiber waveguide to the coupling waveguide is achieved while the taper length can be reduced by 35% compared to conventional two-section linear taper couplers. The taper design algorithm is further optimized to make the adiabatic taper couplers tolerant to variations in incident light polarization, operation wavelength, and dimensional control during fabrication. Finally, we propose and design a taper that adiabatically couples light from the fundamental mode to the first-order mode. Such a taper coupler is useful in an integrated semiconductor optical amplifier/p-i-n detector circuit.