Cosmic rays are ubiquitous in space, and are apparently present wherever the matter density is small enough that they are not removed by collisions with ambient particles. The essential similarity of their energy spectra in many different regions places significant general constraints on the mechanisms for their acceleration and confinement. Diffusive shock acceleration is at present the most successful acceleration mechanism proposed, and, together with transport in Kolmogorov turbulence, can account for the universal specta. In comparison to shock acceleration, statistical acceleration, invoked in many situations, has significant disadvantages. The basic physics of acceleration and transport are discussed, and examples shown where it apparently works very well. However, there are now well-established situations where diffusive shock acceleration cannot be the accelerator. This problem will be discussed and possible acceleration mechanism evaluated. Statistical acceleration in these places is possible. In addition, a new mechanism, called diffusive compression acceleration, will be discussed and shown to be an attractive candidate. It has similarities with both statistical acceleration and shock acceleration.