We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the number of queries of quantum search algorithms, their error probability, the size of the search space, and the number of solutions in this space. Using this, we deduce new lower and upper bounds for quantum versions of amplification problems. Next, we establish nearly optimal quantum-classical separations for the query complexity of monotone functions in the zero-error model (where our quantum zero-error model is defined so as to be robust when the quantum gates are noisy). Also, we present a communication complexity problem related to a total function for which there is a quantum-classical communication complexity gap in the zero-error model. Finally, we prove separations for monotone graph properties in the zero-error and other error models which imply that the evasiveness conjecture for such properties does not hold for quantum computers.