This book is dedicated to Dennis Sullivan on the occasion of his 60th birthday. The framework of affine and hyperbolic laminations provides a unifying foundation for many aspects of conformal dynamics and hyperbolic geometry. The central objects of this approach are an affine Riemann surface lamination $mathcal A$ and the associated hyperbolic 3-lamination $mathcal H$ endowed with an action of a discrete group of isomorphisms. This action is properly discontinuous on $mathcal H$, which allows one to pass to the quotient hyperbolic lamination $mathcal M$.Our work explores natural 'geometric' measures on these laminations. We begin with a brief self-contained introduction to the measure theory on laminations by discussing the relationship between leafwise, transverse and global measures. The central themes of our study are: leafwise and transverse 'conformal streams' on an affine lamination $mathcal A$ (analogues of the Patterson-Sullivan conformal measures for Kleinian groups), harmonic and invariant measures on the corresponding hyperbolic lamination $mathcal H$, the 'Anosov-Sinai cocycle', the corresponding 'basic cohomology class' on $mathcal A$ (which provides an obstruction to flatness), and the Busemann cocycle on $mathcal H$.A number of related geometric objects on laminations - in particular, the backward and forward Poincare series and the associated critical exponents, the curvature forms and the Euler class, currents and transverse invariant measures, $lambda$-harmonic functions and the leafwise Brownian motion - are discussed along the lines. The main examples are provided by the laminations arising from the Kleinian and the rational dynamics. In the former case, $mathcal M$ is a sublamination of the unit tangent bundle of a hyperbolic 3-manifold, its transversals can be identified with the limit set of the Kleinian group, and we show how the classical theory of Patterson-Sullivan measures can be recast in terms of our general approach. In the latter case, the laminations were recently constructed by Lyubich and Minsky in [LM97].Assuming that they are locally compact, we construct a transverse $delta$-conformal stream on $mathcal A$ and the corresponding $lambda$-harmonic measure on $mathcal M$, where $lambda=delta(delta-2)$. We prove that the exponent $delta$ of the stream does not exceed 2 and that the affine laminations are never flat except for several explicit special cases (rational functions with parabolic Thurston orbifold).