Rotordynamics is an important area in mechanical engineering. Many machines contain rotating parts. It is well known that rotating components can develop large amplitude lateral vibrations near certain speeds called critical speeds. This large amplitude vibration is called rotor whirl. This thesis is about rotor whirl. Conventional treatments in rotordynamics use what are called gyroscopic terms and treat the rotor as a one-dimensional structure (Euler-Bernoulli or Timoshenko) with or without rigid masses added to them. Gyroscopic terms are macroscopic inertial terms that arise due to tilting of spinning cross-sections. This approach, while applicable to a large class of industrially important rotors, is not applicable to a general rotor geometry. In this thesis we develop a genuine continuum level three dimensional formulation for rotordynamics that can be used for many arbitrarily shaped rotors. The key insight that guides our formulation is that gyroscopic terms are macroscopic manifestations of the prestress induced due to spin of the rotor. Using this insight, we develop two modal projection techniques for calculating the critical speed of arbitrarily shaped rotors. These techniques along with our prestress based formulation are the primary contributions of the thesis. In addition, we also present two different nonlinear finite element based implementations of our formulation. One is a laborious load-stepping based calculation performed using ANSYS (a commercially available finite element package). The other uses our nonlinear finite element code. The latter two techniques are primarily developed to provide us with an accurate answer for comparison with the results obtained using the modal projection methods. Having developed our formulation and the subsequent modal projection approximations, we proceed to validation. First, we analytically study several examples whose solutions can be easily obtained using routine methods. Second, we consider the problem of a rotating cylinder under axial loads. We use a semi-analytical approach for this problem and offer some insights into the role played by the chosen kinematics for our virtual work calculations. The excellent match with known results obtained using Timoshenko theory validates the accuracy of our formulation. Third, we consider several rotors of arbitrary shape in numerical examples and show that our modal projection methods accurately estimate the critical speeds of these rotors. After validation, we consider efficiency. For axisymmetric rotor geometries, we implement our formulation using harmonic elements. This reduces the dimension of our problem from three to two and considerable savings in time are obtained. Finally, we apply our formulation to describe asynchronous whirl and internal viscous damping phenomena in rotors.