A rollup executes transactions somewhere other than the base chain and posts the results back to it. The claim it makes is that users retain the base chain's security guarantees despite not transacting there. That claim rests on two things being true: the data needed to reconstruct the rollup's state must be published where anyone can get it, and there must be a mechanism by which a wrong result posted to the base chain can be rejected.
Optimistic rollups take the second requirement as a challenge game. The operator posts a state root and asserts it is correct. For a challenge window — conventionally about a week — anyone may submit a fraud proof demonstrating that the posted state does not follow from the published transactions. If nobody does, the root stands. The security assumption is that at least one honest party is watching and able to submit a proof, and the week-long window is why withdrawals from these systems take a week unless a third party fronts the funds.
Zero-knowledge rollups replace the game with a proof. The operator produces a succinct cryptographic proof that the new state follows correctly from the previous state and the batch of transactions, and the base chain verifies it. There is nothing to challenge because the proof establishes correctness directly, so withdrawals are limited by proving time rather than by a dispute period. The cost is complexity: the prover is expensive, and building one for a general-purpose virtual machine is substantially harder than for a fixed application.
Data availability is the requirement people underestimate. If the transaction data is not published, users cannot reconstruct their own balances, cannot generate the proofs needed to exit, and are dependent on the operator's goodwill — which is precisely the position the design exists to avoid. This is the line between a rollup and a validium: a validium keeps proofs on-chain and data off it, which is a defensible engineering trade and a materially weaker guarantee. It is also why Ethereum's introduction of blob space, a cheaper data channel that is not retained forever, mattered so much to rollup economics.
The honest caveats are about maturity rather than architecture. Most production rollups still run a single sequencer, which orders transactions and can censor or reorder them even where it cannot steal. Most retain upgrade keys held by a multisignature wallet, which means the contracts governing user funds can be changed by a small group. And several optimistic systems have operated with permissioned proposer sets, so the honest-watcher assumption depended on a list rather than on anyone. Independent trackers publish these properties per network, and they are the specifics worth checking before treating a rollup's security as equivalent to its base chain's.
The direction of travel is towards shared sequencing, permissionless proving and time-locked upgrades that give users an exit window before a change takes effect. Where a given network sits on that path is a more useful question than which proof system it uses.