The  Zeeman effect and dust grain alignment are two major methods for  probing magnetic fields (B-fields) in molecular clouds, largely  motivated by the study of star formation, as the B-field may regulate  gravitational contraction and channel turbulence velocity. I will review  our observations of B-fields over the past decade, along with our  interpretation. Galactic B-fields anchor molecular clouds down to cloud  cores with scales around 0.1 pc and densities of 10^4–5 H₂/cc.  Within the cores, turbulence can be slightly super-Alfvénic, while the  bulk volumes of parental clouds are sub-Alfvénic. The consequences of  these largely ordered cloud B-fields on fragmentation and star formation  are observed. The above paradigm is very different from the generally  accepted theory during the first decade of the century, when cloud  turbulence was assumed to be highly super-Alfvénic. Thus, turbulence  anisotropy and turbulence-induced ambipolar diffusion are also  revisited.