Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which is the infinite lattice of translation-invariant total orders under containment of inversion sets. The affine Tamari lattice is a quotient of the Dyer lattice, which in turn is a quotient of the cyclic Dyer lattice and is isomorphic to the collection of biclosed sets of the root system of type $\widetilde{A}_{n-1}$ under inclusion. We provide numerous combinatorial and algebraic descriptions of these lattices using translation-invariant total orders, translation-invariant binary in-ordered trees, noncrossing arc diagrams, torsion classes, triangulations, and translation-invariant noncrossing partitions. The cardinalities of the cyclic and affine Tamari lattices are the Catalan numbers of types $B_n$ and $D_n$, respectively. We show that these lattices are self-dual and semidistributive, and we describe their decompositions coming from the Fundamental Theorem of Finite Semidistributive Lattices. We also show that the rowmotion operators on these lattices have well-behaved orbit structures, which we describe via the cyclic sieving phenomenon. Our new combinatorial framework allows us to prove that the lengths of maximal green sequences for the completed path algebra of the oriented $n$-cycle are precisely the integers in the interval $[2n-1,\binom{n+1}{2}]$.