In Canada, we use pencil and paper voting. I am not 100% certain of the very precise details of how ballots are stored and handled prior to voting, but I do know the procedures are similar as for currency.
When you are handed your ballot, a serial-numbered strip called the counterfoil is torn off the ballot paper (there is of course no serial number on the ballot itself), so there is a tally of how many people voted.
The amount of ballots used at each polling table (covers about one city block or about 200-400 people) must be equal to the number of people voted.
Each polling table is staffed by two people, who are either drawn from lists provided by the parties who won and came in second in that electoral district in the previous election, or who are hired by Elections Canada directly. Also, any candidate may designate scrutineers for any or all of the tables, who may monitor the voting process and the count.
The ballots are counted on election night, but are retained in case a recount, conducted by a judge, becomes necessary. (This election, a seat on Montreal’s South Shore actually changed hands due to a recount.)
We do have our little problems (there were rather draconian ID requirements this year that many feel prevented lots of people from voting, especially the poor, rural people, Native people, and young people, and there were also problems with under-trained and in some cases unfit poll officials) but in general there is a high degree of confidence in the system.
Elections Canada is a non-partisan organization and conducts federal elections and referenda throughout Canada.