What about the out-of-pocket expenses of Mesothelioma litigation? Who fronts those expenses?

Each lawyer decides independently whether or not s/he will advance the expenses on your case, subject to his or her state's law. The lawyer's decision may depend on the nature and strength of your case, who the other party is, whether they have adequate insurance, and whether the lawyer has reason to expect that you will follow through with the matter, and not change your mind.

Expenses can include such things as court filing fees, the costs of obtaining records, transcripts of testimony, the costs of expert witnesses, investigative services and travel expenses. They can mount up. Generally the expenses will be deducted from your recovery if your case is successful.

By the way, your contingency fee may be calculated on the recovery before or after expenses, so ask. In some states the law requires you to repay the expenses out-of-pocket if you are unsuccessful.