polymathy

Dawkins, Ancestor's Tale, 2004, says Neanderthals did not interbreed with humans, based on mitochondrial DNA
But Science article, 2010, "a draft sequence of the neanderthal genome" says otherwise. Score!

Fossils indicate that modern anatomy passed to the rest of the world via young-out-of-Africa migrations.

Homo sapiens sapiens reached anatomical modernity 200,000 years ago.
The "young" migration was 50,000 years ago (out of the middle east, apparently)
But other peoples already living there from earlier homo migrations, maybe even homo erectus migrations earlier

erectus migration 1.7 million years ago

homo erectus
homo habilis
homo ergaster
homo heidelbergensis
australopithecus

Dawkins uses erect, archaic, and modern
archaic forms persisted along modern forms until 100,000 years ago (longer if you include the neanderthals)
disputes over terminology and classification
here's one:
homo sapiens heidelbergensis (archaic) gives rise to homo sapiens sapiens (modern)
either homo sapiens neanderthalensis or homo neanderthalensis (depending on whether you want to call them modern or archaic)
little divergence from sapiens sapiens, but anatomically somewhat closer to the archaics, so some call sapiens, some not

First neanderthal fossils appear 130,000 years ago, evolved in Europe & Middle East, disappeared 28,000 years ago

Homo ergaster is the ancestor of heidelbergensis, neanderthalensis, and sapiens
homo erectus may be the same as ergaster, or Asian version of African ergaster
1.3 to 1.8 million years ago
erectus originated in Africa, spread to India, China, and Java

habilis either the direct ancestor, or sharing a close ancestor with, homo ergaster

habilis is the earliest homo
overlaps with ergaster
may be separate lineages

Denisovans share recent ancestor with neanderthals
interbred with humans, lived in Siberia and southeast Asia
Melanesians and Australian aborigines get up to 6% of their DNA from the Denisovans

1.8 million years ago, homo erectus
800,000 years ago, homo antecessor
600,000 years ago homo heidelbergensis, which split into neanderthalensis
125,000 years ago homo sapiens sapiens, into Middle East
50,000 years ago, homo sapiens sapiens, from Middle East to south Asia, Europe 43,000 years ago

Templeton, "Out of Africa Again and Again", Nature 2002
"out of Africa replacement model": Homo erectus in Africa, Europe, and Asia had little contact, homo sapiens sapiens evolved only in Africa, then spread out and replaced everybody
"multiregional trellis model": genetic contact between African and non-African homo erectus maintained, although limited by barriers to gene flow
human lineage started in Africa and spread out of Africa 1.7 million ya
little evidence for recurrent gene flow among the Homo erectus populations shortly after they came out of Africa
recurrent gene flow did occur back to at least 600,000 years ago and possibly more
mtDNA and Y-DNA tightly clustered around an expansion out of Africa event ~100,000 years ago
another expansion out of Africa, which was not a replacement event, around 80,000 to 150,000 years ago
a replacement event would have erased the evidence of the mt-DNA and Y-DNA bottleneck

1.7 million years ago, homo erectus to Eurasia, fossil dating
little evidence of recurrent gene flow immediately after
increasing evidence of recurrent gene flow as time progresses
420,000 to 840,000 years ago interbreeding, not replacement event
80,000 to 150,000 years ago interbreeding, not replacement event, mtDNA and Y-DNA tightly clustered around this event
Y-DNA and beta-globin detect a recent out-of-Asia event, that is not shown by mtDNA at all, and therefore was male-driven and consisted of interbreeding, not replacement
genetic contact among human populations after most recent out of Africa event is indicated
only Americas are significantly isolated

African and Eurasian populations were linked by recurrent gene flow, certainly over the last half a million years, and probably longer
occasional major movements involving interbreeding
at least two major movements of peoples out of Africa after the initial spread of homo erectus 1.7 million years ago
one I take to be heidelbergensis, and the other homo sapiens sapiens
90% of the haplotype trees are rooted in Africa
total replacement is falsified, which means all haplotype trees greater than 100,000 years ago should be rooted in Africa
recent out of Asia was more recent than homo sapiens

There are many models. Here is one based on what I've been reading and which takes into account recent findings.

According to the fossil record, homo erectus spread out of Africa into Eurasia 1.7 million years ago
There continued to be gene flow, restricted by distance, among the different populations
Somewhere after this, the ancestors of Denisovans left Africa, entered into Asia
600,000 years ago (roughly), homo heidelbergensis left Africa, interbred with the local populations and did not totally replace them
400,000 years ago, Neanderthals evolved from heidelbergensis in the Middle East and Europe
200,000 years ago (or later, per Templeton?), homo sapiens sapiens achieved anatomical modernity in Africa
125,000-80,000 years ago, homo sapiens sapiens left Africa for the Middle East. They there interbred with Neanderthals, based on Neandertals being closer to non-Africans than to Africans, but with no differing degrees of similarity to French, Han Chinese, or Papuan individuals.
50,000 years ago, homo sapiens sapiens spreads into Asia
somewhere in Asia, probably in southeast Asia, Denisovans interbred with the ancestors of Melanesians, Australians
43,000 years ago, homo sapiens sapiens spreads into Europe
I'm not sure we could tell if Neanderthals contributed to human DNA in Europe, given that the expansion of humans into the Americas cannot be accurately dated by phylogenetic analysis
mtDNA can tell there was an Amerindian split (but not when it was?), and there's no evidence from mtDNA of Neandertal interbreeding?