CMOS SRAM area scaling is slowing down due to several challenges faced by transistors at nanoscale such as increased leakage. This calls for new concepts and technologies to overcome CMOS scaling limitations. In this thesis, we propose a multi-state memory to store multiple bits in a single cell, enabled by graphene and graphene nanoribbon crossbar devices (xGNR). This could provide a new dimension for scaling. We present a new multi-state volatile memory fabric called Graphene Nanoribbon Tunneling Random Access Memory (GNTRAM) featuring a heterogeneous integration between graphene and CMOS. A latch based on the xGNR devices is used as the memory element which exhibits 3 stable states. We propose binary and ternary GNTRAM and compare them with respect to 16nm CMOS SRAM and 3T DRAM. Ternary GNTRAM (1.58 bits/cell) shows up to 1.77x density-per-bit benefit over CMOS SRAMs and 1.42x benefit over 3T DRAM in 16nm technology node. Ternary GNTRAM is also up to 1196x more power-efficient per bit against high-performance CMOS SRAMs during stand-by. To enable further scaling, we explore two approaches to increase the number of bits per cell. We propose quaternary GNTRAM (2 bits/cell) using these approaches and extensively benchmark these designs. The first uses additional xGNR devices in the latch to achieve 4 stable states and the quaternary memory shows up to 2.27x density benefit vs. 16nm CMOS SRAMs and 1.8x vs. 3T DRAM. It has comparable read performance in addition to being power-efficient, up to 1.32x during active period and 818x during stand-by against high performance SRAMs. However, the need for relatively high-voltage operation may ultimately limit this scaling approach. An alternative approach is also explored by increasing the stub length in the xGNR devices, which allows for storing 2 bits per cell without requiring an increased operating voltage. This approach for quaternary GNTRAM shows higher benefits in terms of power, specifically up to 4.67x in terms of active power and 3498x during stand-by against high-performance SRAMs. Multi-bit GNTRAM has the potential to realize high-density low-power nanoscale memories. Further improvements may be possible by using graphene more extensively, as graphene transistors become available in future.