We formalize the classical complexity classes $\mathrm{L}$, $\mathrm{NL}$, $\mathrm{P}$, $\mathrm{NP}$, $\mathrm{coNL}$, $\mathrm{coNP}$, $\mathrm{PSPACE}$, $\mathrm{NPSPACE}$, and $\mathrm{EXPTIME}$ for languages of finite binary strings. P uses deterministic stack machines, identity encoding of binary inputs, and singleton Boolean outputs. NP uses polynomially bounded certificates and polynomial-time verification. We prove equivalence with finite single-tape machines and closure under complement. EXPTIME uses these single-tape machines. The space classes use finite Turing machines with a bounded, read-only input tape and a separate work tape, whose space is bounded on every computation branch. We prove the full chain through $\mathrm{L}\subseteq\mathrm{NL}$, $\mathrm{NL}\subseteq\mathrm P$, $\mathrm{P}\subseteq\mathrm{NP}$, $\mathrm{NP}\subseteq\mathrm{PSPACE}$, $\mathrm{PSPACE}=\mathrm{NPSPACE}$ (Savitch's theorem), and $\mathrm{NPSPACE}\subseteq\mathrm{EXPTIME}$. We also state $\mathrm{P}\ne\mathrm{NP}$ as an open question.