Sooner or later, your revocable living trust will become irrevocable. Usually, it happens when you die: at that point, neither you nor anyone else can change the trust terms. If you made yourself the original trustee to keep control of the trust assets, then control of the trust passes at your death to your designated successor trustee. But if something happens to your trustee, your living trust will hopefully specify what to do next.
When the trustee of a revocable trust dies, the successor trustee takes over. If no successor was named, the trust could end up in probate court.
When you draw up the declaration of trust -- the paperwork that establishes your revocable trust -- you typically identify yourself as both grantor and trustee, and name the successor trustee. At your death, she takes over and carries out the instructions you wrote in the declaration. Typically, these direct her to distribute the trust assets to beneficiaries, or to manage them for children until they come of age. When she distributes the last assets, the trust dissolves.
You and your partner can put property into a shared living trust together. Usually, couples who do this serve as joint trustees and as beneficiaries. If your partner dies, you become sole trustee. When you die, the successor trustee takes over. The trust doesn't become irrevocable until you both die, so you can change or revoke the trust after your partner's death.
A trust has to have a trustee. If your successor trustee refuses to serve after you die, or if she dies first and you don't pick a new trustee, the probate court will step in. In California, for example, the court follows any recommendations you make in the declaration for choosing a trustee. If you made none, the judge looks for a trust company acceptable to the beneficiaries. If that doesn't work, the judge appoints someone. If your beneficiaries have a recommendation for the nominee, the judge may consider it.
When you die, your successor has her work cut out for her. She has to learn which assets she controls, how much they're worth and how you want them disposed of. She must close trust accounts and distribute the proceeds, and transfer title to houses or other property. With a good-sized estate, it's a demanding job, and one with opportunities for dishonest self-enrichment. Choose your trustee carefully, and ask her if she's willing to do the job before you appoint her in the declaration. It doesn't hurt to name a couple of successors to your successor too.