Casimir forces between macroscopic objects arise due to quantum fluctuations of the electromagnetic field. In addition to being theoretically interesting, recent experiments also indicate that Casimir forces (usually attractive and increasing with decreasing object separation) may contribute significantly to friction in microelectromechanical systems (MEMS). In order to gain a better understanding of this effect, scientists have begun developing new theoretical methods capable of computing the force in complex geometries (beyond simple parallel-plate geometries or similar approximations thereof ).In this talk, I will explore a new approach for computing Casimir forces based on the finite-difference time-domain method (FDTD) that is applicable to complex three-dimensional geometries, and which has recently been exploited to study forces in geometries exhibiting unusual effects, such as repulsion, stable suspension and levitation. Our approach allows researchers to exploit powerful, widely available and parallel FDTD software that can handle large three-dimensional geometries, anisotropic dielectrics and even periodic media, without requiring any software modification. Because Casimir forces are most easily computed in the imaginary-frequency domain, formulating the equivalent problem in the time domain requires some care, and in particular, the imaginary-time evolution of Maxwell’s equations gives rise to undesirable, exponentially growing fields. We show how it is possible to circumvent this problem by exploiting a correspondence (derived here) between complex-frequency contour deformations and material deformations. The resulting FDTD algorithm involves the evolution of Maxwell’s equations in real time but with a modified vacuum, corresponding to a constant DC conductive fluid. I will end the talk by giving a brief survey of some of the interesting effects that can be studied using these general methods.