Recent impactful advances in integrated photonics undoubtedly owe much to silicon and its associated enabling platform (SOI). Although silicon has proved to be an indispensable element in many photonic systems yet it seems that it is not the ultimate solution to address all the challenges facing the photonics community. Therefore, integration of silicon with other optical materials featuring diverse properties is highly desirable. Such integration will be conducive to platforms which are naturally more capable and are suited for implementation of a wider range of optical devices and diverse functionalities. This dissertation is dedicated to design and implementation of integrated optical elements for hybrid material platforms. The basic theoretical foundation of integrated photonics is laid out in Chapter 2. In Chapter 3, an interlayer grating coupler for a specific hybrid material platform is designed, and demonstrated. Considering the fact that in almost all integrated photonic platforms, fabrication imperfections lead to an unpredictable shift in the wavelength of operation of individual devices, post fabrication tuning/trimming is inevitable. A number of widely used post fabrication trimming/tuning methods are briefly reviewed in Chapter 4 with special emphasis on a method based on electron beam exposure. In Chapter 5, an ultra-fast, low-power, and self-trimmable electro-optic modulator in demonstrated on a Si-based multilayer platform. Due to its remarkable optical and electronic properties, graphene has become a valuable material for opto-electronic applications. Integration of this novel 2D material with SOI platform is investigated in Chapter 6. Graphene-based electro-optic modulation through absorption and refractive-index change is successfully demonstrated using electrostatic gating mechanism. Chapter 7 is devoted to demonstration of a field-programmable 2 by 2 optical switch on a vertically stacked Si/SiO2/SOI platform. In Chapter 8, the peak-dragging phenomenon in a nanobeam photonic crystal cavity is studied. The optical bistability associated with this nonlinear phenomenon is of great interest for all-optical processing and sensing application. Future directions of this thesis are also discussed in the last Chapter.