College Answer

Coronary artery bypass grafting off bypass is now widely performed. There are numerous techniques for getting access to the grafts and heart (eg. limited thoracotomy, video assisted).

(a) Theoretical advantages include: 
•  Avoidance of  the effects of extracorporeal circulation including .. coagulation/kalikrein activation, microembolus of air and platelet clumps to cerebral circulation 
•  Avoidance of aortic cannulation with attendant risks of  arterial  embolisation of  air or atheroma 
•    Avoidance of atrial cannulation with attendant risk of atrial injury respectively 
•    Avoidance of the effects of cardioplegia including K load. fluid load, coronary air embolus 
•    Avoidance  of  risks  of  aortic  cross  clamping  including  atheroembolism. myocardial ischaemia
•  Decreased costs (less equipment, less staff)

(b) Theoretical disadvantages include: 
•  Potential for  myocardial ischaemia without the  protection of  cold cardioplegia during grafting. 
•  Unfavorable operating conditions with a beating heart increasing the risk of anastamotic bleeding, suboptimal revascularisation and myocardial ischaemia

[Information on graft patency, morbidity, mortality and quality of life from proper trials is not yet available]

Current status and future directions of minimally invasive cardiac surgery. Current Opinion in Cardiology 1999,14(5); 419·

Minimally invasive coronary bypass: a dissenting opinion. Circulation 98(6);495-7

Discussion

There is little to add to the college answer. It is rare for references to be included with it, and I will link to these papers below. The second one is available for free as full text, from Circulation.

I will quote from it:

"... just because something can be done by some surgeons does not mean that it should be done by all surgeons. "

A comparison of conventional and off-bypass CABG appears as a brief summary in the Required reading section. Additionally, LITFL have this article ("Cardiothoracic Surgery Literature Summaries") where two of the more influential on-vs-off-pump trials are thoroughly dissected. Put simply, the advantages and disadvantages are as follows:

Advantages of off-pump CABG:

• Protection from circuit-related complications:
• Avoidance of aortic cannulation
• Avoidance of atrial cannulation
• Avoidance of the effects of cardioplegia
• Avoidance  of  risks  of  aortic  cross  clamping
• Decreased costs (less equipment, less staff)
• A meta-analysis in 2012 concluded that off-pump bypass "reduces the incidence of post-operative stroke by 30% and has no notable effect on mortality or myocardial infarction"

Disadvantages of off-pump CABG:

• Potential for  myocardial ischaemia without the  protection of  cold cardioplegia during grafting.
• Unfavorable operating conditions with a beating heart increasing the risk of anastamotic bleeding, suboptimal revascularisation and myocardial ischaemia
• A recent meta-analysis has demonstrated that with off-pump grafts, there is an increased risk of graft occlusion with all grafts except LIMA and radial artery grafts.