We develop a surface theory in pseudohermitian geometry. We define a notion of (p-)mean curvature and the associated (p-)minimal surfaces. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the p-mean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining function, and as a quantity in terms of calibration geometry. We also show that there are no closed, connected, $C^{2}$ smoothly embedded constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.