What do Christians mean by "Jesus took our illnesses upon himself on the cross" and why? Having a bit of holiday over Christmas and New Year I'm going over some notes I've made of topics I want to look at.

One such question regards the use of terminology such as "Jesus took our illnesses upon himself on the cross".

I've read a little and seen that there's a variety of perspectives (from perhaps "sickness is something that shouldn't affect Christians" to perhaps a focus on the full-application being in the future kingdom) that this statement could cover, but I'd be interested to see if anyone could give me the background texts to support such a statement (one would not have to agree with the exegesis to present the texts used to support the doctrine).

One commonly referenced text would be Isa 53:4-5 which concludes


  and with his wounds we are healed


I'm interested in seeing what is the range of ideas that "Jesus took our illnesses" covers and what key texts are used to build up that argument.

I hope that's reasonably clear!
In Isa 53, the word you are translating "illnesses" is more often translated as "transgressions" or "iniquities."  Both of those words highlight the fact that Jesus took our sins with him on the cross, and by the wounds he suffered, we were healed - from the iniquity of sin.

The basic idea of salvation in Christainity is often summarized as this -


We are sinners
Because of this sin, we deserve eternal death - separation from God
Jesus paid the penalty for that sin by dying on the cross, thus we are healed and given eternal life.


(For the purposes of this answer, lets not get into all the different ideas of atonement, etc... I'm going for the overview here!)

The point is "illness" is not a physical thing here, but a legal one. (Indeed, I've never seen that translation before - perhaps its just being confused with iniquities?  An iniquity is an evil, a cause of suffering and distress, not necessarily a bodily thing.)  We are "suffering" from sin, and the Suffering Servant of Isaiah 53 pays for that sin, so we are healed.

Beyond that, I've never seen a promise that Christians would never get sick.  Even the early church recognized that believers would still die.  The Good News, however, is that in our resurrected state, there would be no more corruption in our bodies - i.e. no sickness - but that's on the other side of the Jordan, not this one!