Brain Imaging Studies
Brain imaging studies have been done with adult stutterers but not with children who stutter. We don’t know whether the neurological abnormalities found in adult stutterers cause stuttering or were caused by stuttering, i.e., whether growing up stuttering causes a normal child’s brain to develop abnormalities.
Different brain imaging studies found different results, partly because different technology was used, and partly because different speaking tasks were tested. E.g., the production of sentences looks different from the production of single words.
A 1996 positron emission tomography (PET) study found
…overactivation of motor system areas in both cerebrum and cerebellum. During stuttering, there was also relative deactivation of left hemisphere auditory areas. These areas are thought to support self-monitoring of speech production. Another area typically activated in speech production tasks in the frontal temporal regions was similarly deactivated. During choral reading, which induced fluent speech production, these atypical patterns of over- and under-activation were largely eliminated. The authors speculated that stuttering results from over-activation of the speech motor system and a deficiency in self-monitoring of speech production. 
Several other studies also found associated stuttering with underactivation in auditory perception and processing areas. One study found that the impaired area integrates how we hear our voices (auditory perception) and how we feel our muscles moving (somatic perception). 
Speech is normally produced in the left hemisphere, particularly in the Broca’s area. Two functional magnetic resonance imaging (fMRI) studies found abnormal activation in the right frontal operculum (RFO). This area is associated with time estimation tasks, and is selectively recruited when speech tasks become more complex. The RFO is considered to be the right hemisphere’s equivalent of Broca’s area, which is area associated with speech production. 
Several studies found that patterns of activity areas normally seen during speech in non-stutterers’ brains “were either absent, bilateral in nature, or lateralized to the right hemisphere.”  It’s not clear whether this is because something is wrong with certain left hemisphere areas in stutterers, or if something else is going on, such as stutterers experiencing right-hemisphere emotional activity during simple speaking tasks that don’t elicit emotional activation in non-stutterers.
A magnetoencephalography (MEG) study, which can track sequences of events, found that the normal sequence that begins with programming articulatory gestures and then goes to motor preparation of these gestures was reversed in stutterers. 
The dopamine abnormalities associated with stuttering are in the article Genetics of Stuttering.
Changes Following Stuttering Therapy
Several studies have followed adult male stutterers before and after completing a prolonged speech (fluency shaping) stuttering therapy program. In one study stuttered words were reduced from 7% to 1.6%. Before therapy the stutterers’ brains had a pattern of abnormal over-activation in speech motor control areas and in right hemisphere areas. After therapy activation of these areas decreased and moved to the left hemisphere. These changes were also seen one year post-therapy. 
Abnormal asymmetry has been found between the left and right planum temporale, when stutterers were compared to non-stutterers. Another study found anatomical differences in the Rolandic operculum, which is close to the area that controls oral function, and in the arcuate fasciculus, which links Wernicke’s area (language production) to Broca’s area (speech production). 
 Braun, A.R., Varga, M., Stager, S., Schulz, G., Selbie, S., Maisog, J.M., Carsom, R.E., & Ludlow, C.L. (1997). Atypical Lateralization of Hemispehral Activity in Developmental Stuttering: An H215O Positron Emission Tomography Study. In Speech Production: Motor Control, Brain Research and Fluency Disorders, edited by W. Hulstijn, H.F.M. Peters, and P.H.H.M. Van Lieshout, Amsterdam: Elsevier.