Mate in three or more is lesson 6 of 9 in Real-game tactics 9: Master calculation, a tactics course suited to players rated around 2200 (advanced players). Longer mating combinations: keep checking, keep the king boxed in, and count every reply. It has 5 interactive exercises: 5 calculated lines from real positions. You play every move on a board and get feedback straight away. The lesson is free.
What you practise
Black to move: mate in 4.
Black to move: mate in 3.
White to move: mate in 5.
Black to move: mate in 5.
White to move: mate in 4.
Worked examples
Black to move: mate in 4. 1...Ra1+ 2.Kb3 Na5+ 3.Kb4 Qc4+ 4.Kxa5 Qb5#. Every move in the line was forcing. Mate in 4.
Black to move: mate in 3. 1...Qh3 2.Qf1 Nxf1 3.Rxf1 Qg2#. Taking the queen on f1 removed the defender of g2. Mate in 3.