This national shrine is certainly one of Montreal's most famous landmarks. It is a beautiful building outside and visible from all over Montreal as it sits on its hill and then is perched above many stairs. For this reason it can be difficult for people who have trouble climbing stairs, although you can drive up relatively close (though you can't park up there) and then enter through a basement area and take an elevator. Once inside there are escalators to take you up even further (and elevators). During Lent you might notice people crawling (literally) on the center stairs as they penitently make their way up to a church service while praying.
The view from the balconies outside are amazing, it is definitely worth it to go up there just to see Montreal from that height. One of the things you'll be able to see is the giant "Orange" by Namur metro station: it's an orange julep fast food place, also worth visiting, and funny to see from that height.
The interior is very interesting, although personally I found it cold and somewhat austere. There is an incredibly massive chandelier, and some beautiful stained glass. But you will notice that it is relatively modern looking, having been completed only in 1967.
Another fascinating feature is that Brother Andre's heart is stored in the reliquary, it is attributed to have healed people, as he did when he was alive (and founded St. Joseph as a chapel in the early 1900's). Over 2 million or so pilgrims come to visit the relic every year.
This basilica has great meaning and beauty for non-religious and religious people alike, it is certainly worth a visit if you are in Montreal for more than a day or so.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC