The multi-net Global Routing Problem (GRP) in VLSI physical design is a problem of routing a set of nets subject to limited resources and delay constraints. Various state-of-the-art routers are available but their main focus is to optimize the wire length and minimize the over ow. However optimizing wire length do not necessarily meet timing constraints at the sink nodes. Also, in modern nano-meter scale VLSI process the consideration of process variations is a necessity for ensuring reasonable yield at the fab. In this work, we try to nd a fundamental strategy to address the timing-driven Steiner tree construction (i.e., the routing) problem subject to congestion constraints and process variation. For congestion mitigation, a gradient based concurrent approach (over all nets) of Erzin et. al., rather than the traditional (sequential) rip-and-reroute is adopted in or- der to propagate the timing/delay-driven property of the Steiner tree candidates. The existing sequential rip-up and reroute methods meet the over ow constraint locally but cannot propagate the timing constraint which is non-local in nature. We build on this approach to accommodate the variation-aware statistical delay/timing requirements. To further reduce the congestion, the cost function of the tree generation method is updated by adding history based congestion penalty to the base cost (delay). Iterative use of the timing-driven Steiner tree construction method and history based tree construction procedure generate a diverse pool of candidate Steiner trees for each net. The gradient algorithm picks one tree for each net from the pool of trees such that congestion is e ciently controlled. As the technology scales down, process variation makes process dependent param- eters like resistance, capacitance etc non-deterministic. As a result, Statistical Static Timing Analysis or SSTA has replaced the traditional static timing in nano-meter scale VLSI processes. However, this poses a challenge regarding the max/min-plus algebra of Dijkstra like approximation algorithm that builds the Steiner trees. A new approach based on distance between distributions for nding maximum/minimum at the nodes is presented in this thesis. Under this metric, the approximation algorithm for variation aware timing driven congestion constrained routing is shown to be provably tight and one order of magnitude faster than existing approaches (which are not tight) such as the MVERT. The results (mean value) of our variation aware router are quite close to the mean of the several thousand Monte Carlo simulations of the deterministic router, i.e the results converge in mean. Therefore, instead of running so many deterministic Monte Carlo simulations, we can generate an average design with a probability distribution reasonably close to that of the actual behaviour of the design by running the proposed statistical router only once and at a small fraction of the computational e ort involved in physical design in the nano regime VLSI. The above approximation algorithm is extended to local routing, especially non- Manhattan lambda routing which is increasingly being allowed by the recent VLSI tech- nology nodes. Here also, we can meet delay driven constraints better and keep related wire lengths reasonable.