I live and die by my iPhone.
Every birthday has a reminder set for the week before to buy a card and present (if needed)
My phone has all my commitments in and locations so it tells me when I need to leave.
Everything else, I use Siri to set up as a reminder. I just speak into my phone and it's done in 10 secs. The trick is to put in reminders for the day / week before, so you know it is coming!
Everything else I have an app - Christmas present app, shopping list app...
My other saviour is my fridge!
My DC each has a peg on the fridge. All paperwork belonging to each child gets stuck onto their peg - their older, so their responsibility not mine, but when they were younger, I did it. Means I can always find their school letters / party invites / info about hobbies etc etc
Also have a laminated list of what needs doing each day / packed lunches / after school snacks / who is dropping or collecting from school and what kit needs to be taken on what days. Easy checklist for DH and the DC to see what they need so it's not all down to me.
Calendar for each month with just the key events on.
And a mini whiteboard to write down the items that need buying.
Finally, as back up I do have a diary. Mostly for work, but I back up my key events in there. I colour code everything with highlighters - red that day, orange for things that need to be done that week and green for things that I need a bit more time on.
But really, the key to our organisation is our weekly diary meetings. Once a week, I go through the calendar app on my phone, the lists on my fridge and make sure I have got everything in my diary. This is my Sunday night job. DH, the DC and I have a family diary meeting on a Sunday, where we discuss what commitments we have got on, who is doing pick up and drop off, the girls write into their diaries what they need to take to school that week, they hand over slips for signing etc etc... all gets done in 20 mins each week.